New Conjectures about Zeroes
of Riemann’s Zeta Function
Yuri Matiyasevich
St.Petersburg Department
of Steklov Institute of Mathematics
of Russian Academy of Sciences
http://logic.pdmi.ras.ru/~yumat
Blowing the Trumpet
“The physicist George Darwin used to
say that every once in a while one
should do a completely crazy experiment, like blowing the trumpet to
the tulips every morning for a month.
Probably nothing would happen, but
what if it did?”
Riemann’s zeta function
Riemann’s zeta function
Dirichlet series:
ζ(s) =
∞
X
n=1
n−s =
1
1
1
1
+ s + s + ··· + s + ...
1s
2
3
n
Riemann’s zeta function
Dirichlet series:
ζ(s) =
∞
X
n=1
n−s =
1
1
1
1
+ s + s + ··· + s + ...
1s
2
3
n
The series converges for s > 1 and diverges at s = 1:
1 1 1 1
+ + + + ...
1 2 3 4
Basel Problem (Pietro Mengoli, 1644)
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
1
1
1
1
+ 2 + 2 + · · · + 2 + · · · = ζ(2)
2
1
2
3
n
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Leonhard Euler:
1
1
1
1
+ 2 + 2 + · · · + 2 + · · · = ζ(2)
2
1
2
3
n
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Leonhard Euler:
1
1
1
1
+ 2 + 2 + · · · + 2 + · · · = ζ(2) = 1.64493406684822644 . . .
2
1
2
3
n
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Leonhard Euler:
1
1
1
1
+ 2 + 2 + · · · + 2 + · · · = ζ(2) = 1.64493406684822644 . . .
2
1
2
3
n
π2
= 1.64493406684822644 . . .
6
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Leonhard Euler:
1
1
1
1
+ 2 + 2 + · · · + 2 + · · · = ζ(2) = 1.64493406684822644 . . .
2
1
2
3
n
π2
= 1.64493406684822644 . . .
6
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Leonhard Euler:
1
1
1
1
+ 2 + 2 + · · · + 2 + · · · = ζ(2) = 1.64493406684822644 . . .
2
1
2
3
n
π2
= 1.64493406684822644 . . .
6
ζ(2) =
π2
6
Basel Problem (Pietro Mengoli, 1644)
1
1
1
1
+ 2 + 2 + ··· + 2 + ··· = ?
2
1
2
3
n
Leonhard Euler:
1
1
1
1
+ 2 + 2 + · · · + 2 + · · · = ζ(2) = 1.64493406684822644 . . .
2
1
2
3
n
π2
= 1.64493406684822644 . . .
6
ζ(2) =
π2
6
Other values of ζ(s) found by Euler
ζ(2) =
ζ(4) =
ζ(6) =
ζ(8) =
ζ(10) =
ζ(12) =
ζ(14) =
1 2
π
6
1 4
π
90
1 6
π
945
1 8
π
9450
691
π 10
638512875
2
π 12
18243225
3617
π 14
325641566250
Other values of ζ(s) found by Euler
ζ(2) =
ζ(4) =
ζ(6) =
ζ(8) =
ζ(10) =
ζ(12) =
ζ(14) =
1 2
π
6
1 4
π
90
1 6
π
945
1 8
π
9450
691
π 10
638512875
2
π 12
18243225
3617
π 14
325641566250
Bernoulli numbers
Bernoulli numbers
∞
X 1
x
=
Bk x k
ex − 1
k!
k=0
B0 = 1,
B10 =
1
B2 = ,
6
5
,
66
1
B1 = − ,
2
B4 = −
B12 = −
691
,
2730
1
,
30
B6 =
7
B14 = ,
6
1
,
42
B8 = −
B16 = −
1
,
30
3617
510
B3 = B5 = B7 = B9 = B11 = · · · = 0
Other values of ζ(s) found by Euler
ζ(6) =
ζ(8) =
ζ(10) =
ζ(12) =
ζ(14) =
1 6 25
π = B6 π 6
6!
945
27
1 8
π = − B8 π 8
8!
9450
691
29
π 10 =
B10 π 10
638512875
10!
211
2
B12 π 12
π 12 = −
12!
18243225
213
3617
B14 π 14
π 14 =
14!
325641566250
Other values of ζ(s) found by Euler
ζ(6) =
ζ(8) =
ζ(10) =
ζ(12) =
ζ(14) =
1 6 25
π = B6 π 6
6!
945
27
1 8
π = − B8 π 8
8!
9450
691
29
π 10 =
B10 π 10
638512875
10!
211
2
B12 π 12
π 12 = −
12!
18243225
213
3617
B14 π 14
π 14 =
14!
325641566250
ζ(2k) = (−1)k+1
22k−1
B2k π 2k
(2k)!
k = 1, 2, . . .
One more value of ζ(s) found by Euler
One more value of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . .
One more value of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
1
2
One more value of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
(1 − 2 · 2−s )ζ(s)
1
2
One more value of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
1
2
(1 − 2 · 2−s )ζ(s) = (1 − 2 · 2−s )(1−s + 2−s + 3−s + 4−s + . . . )
= 1−s + 2−s + 3−s + 4−s + . . .
− 2 · 2−s
− 2 · 4−s − . . .
= 1−s − 2−s + 3−s − 4−s + . . .
One more value of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
1
2
(1 − 2 · 2−s )ζ(s) = (1 − 2 · 2−s )(1−s + 2−s + 3−s + 4−s + . . . )
= 1−s + 2−s + 3−s + 4−s + . . .
− 2 · 2−s
− 2 · 4−s − . . .
= 1−s − 2−s + 3−s − 4−s + . . .
The alternating Dirichlet series
1−s − 2−s + 3−s − 4−s + . . .
converges for s > 0.
One more value of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
1
2
(1 − 2 · 2−s )ζ(s) = (1 − 2 · 2−s )(1−s + 2−s + 3−s + 4−s + . . . )
= 1−s + 2−s + 3−s + 4−s + . . .
− 2 · 2−s
− 2 · 4−s − . . .
= 1−s − 2−s + 3−s − 4−s + . . .
The alternating Dirichlet series
1−s − 2−s + 3−s − 4−s + . . .
converges for s > 0.
Yet another values of ζ(s) found by Euler
Yet another values of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . .
Yet another values of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
1
2
Yet another values of ζ(s) found by Euler
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
ζ(−1) = 11 + 21 + 31 + · · · = 1 + 2 + 3 + . . .
1
2
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · =
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · = 0
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · = 0
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
ζ(−3) = 13 + 23 + 33 + · · · = 1 + 8 + 27 + . . .
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · = 0
1
ζ(−3) = 13 + 23 + 33 + · · · = 1 + 8 + 27 + . . . =
120
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · = 0
1
ζ(−3) = 13 + 23 + 33 + · · · = 1 + 8 + 27 + . . . =
120
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · = 0
1
ζ(−3) = 13 + 23 + 33 + · · · = 1 + 8 + 27 + . . . =
120
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
ζ(−m) = −
Bm+1
2m + 2
m = 0, 1, . . .
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · = 0
1
ζ(−3) = 13 + 23 + 33 + · · · = 1 + 8 + 27 + . . . =
120
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
ζ(−m) = −
Bm+1
2m + 2
m = 0, 1, . . .
ζ(−2) = ζ(−4) = ζ(−6) = · · · = 0
Yet another values of ζ(s) found by Euler
1
2
1
1
1
1
ζ(−1) = 1 + 2 + 3 + · · · = 1 + 2 + 3 + . . . = −
12
ζ(−2) = 12 + 22 + 32 + · · · = 1 + 4 + 9 + · · · = 0
1
ζ(−3) = 13 + 23 + 33 + · · · = 1 + 8 + 27 + . . . =
120
ζ(0) = 10 + 20 + 30 + · · · = 1 + 1 + 1 + . . . = −
ζ(−m) = −
Bm+1
2m + 2
m = 0, 1, . . .
ζ(−2) = ζ(−4) = ζ(−6) = · · · = 0
The functional equation
The functional equation
ζ(2k) = (−1)k+1
22k−1
B2k π 2k
(2k)!
k = 1, 2, . . .
The functional equation
ζ(2k) = (−1)k+1
ζ(−m) = −
22k−1
B2k π 2k
(2k)!
Bm+1
2m + 2
k = 1, 2, . . .
m = 0, 1, . . .
The functional equation
ζ(2k) = (−1)k+1
ζ(−m) = −
22k−1
B2k π 2k
(2k)!
Bm+1
2m + 2
k = 1, 2, . . .
m = 0, 1, . . .
ζ(1 − 2k) = (−1)k 21−2k π −2k (2k − 1)!ζ(2k)
k = 1, 2, . . .
The functional equation
ζ(2k) = (−1)k+1
ζ(−m) = −
22k−1
B2k π 2k
(2k)!
Bm+1
2m + 2
k = 1, 2, . . .
m = 0, 1, . . .
ζ(1 − 2k) = (−1)k 21−2k π −2k (2k − 1)!ζ(2k)
k = 1, 2, . . .
Riemann’s zeta function
Dirichlet series:
ζ(s) =
P∞
n=1 n
−s
Riemann’s zeta function
Dirichlet series:
ζ(s) =
P∞
n=1 n
s = σ + it
−s
Riemann’s zeta function
6
Dirichlet series:
4
2
ζ(s) =
P∞
n=1
n−s
-2
s = σ + it
5
10
-4
-6
The series converges in the semiplane <(s) > 1
15
20
Riemann’s zeta function
6
Dirichlet series:
4
2
ζ(s) =
P∞
n=1
n−s
-2
s = σ + it
5
10
15
20
-4
-6
The series converges in the semiplane <(s) > 1 and defines a function
that can be analytically extended to the entire complex plane except for
the point s = 1, its only (and simple) pole.
Riemann’s zeta function
Euler’s values of ζ(s) for negative integer s were correct:
ζ(−m) = −
Bm+1
2m + 2
m = 0, 1, . . .
Riemann’s zeta function
Euler’s values of ζ(s) for negative integer s were correct:
ζ(−m) = −
Bm+1
2m + 2
m = 0, 1, . . .
ζ(−2) = ζ(−4) = ζ(−6) = · · · = 0
Riemann’s zeta function
Euler’s values of ζ(s) for negative integer s were correct:
ζ(−m) = −
Bm+1
2m + 2
m = 0, 1, . . .
ζ(−2) = ζ(−4) = ζ(−6) = · · · = 0
Points s1 = −2, s2 = −4, . . . , sk = −2k, . . . are called trivial zeroes
of the zeta function
Riemann’s zeta function
Euler’s values of ζ(s) for negative integer s were correct:
ζ(−m) = −
Bm+1
2m + 2
m = 0, 1, . . .
ζ(−2) = ζ(−4) = ζ(−6) = · · · = 0
Points s1 = −2, s2 = −4, . . . , sk = −2k, . . . are called trivial zeroes
of the zeta function, and it has no other real zeroes.
The functional equation
Euler:
ζ(1 − 2k) = (−1)k 21−2k π −2k (2k − 1)!ζ(2k)
k = 1, 2, . . .
The functional equation
Euler:
ζ(1 − 2k) = (−1)k 21−2k π −2k (2k − 1)!ζ(2k)
k = 1, 2, . . .
Riemann:
ζ(1 − s) = cos
πs 2
21−s π −s Γ(s)ζ(s)
s = σ + it
Function ξ(s)
ζ(1 − s) = cos
πs 2
21−s π −s Γ(s)ζ(s)
Function ξ(s)
ζ(1 − s) = cos
cos
πs 2
=
πs π
Γ(− 2s
+
1
s
2 )Γ( 2
+ 21 )
2
21−s π −s Γ(s)ζ(s)
Function ξ(s)
ζ(1 − s) = cos
cos
πs 2
=
πs π
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ
Γ(s) = √ Γ
2
π
s
1
+
2 2
Function ξ(s)
ζ(1 − s) = cos
cos
πs 2
=
πs π
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ
Γ(s) = √ Γ
2
π
s
1
+
2 2
Function ξ(s)
ζ(1 − s) = cos
cos
πs 2
=
πs π
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ
Γ(s) = √ Γ
2
π
s
1
+
2 2
Function ξ(s)
ζ(1 − s) = cos
cos
π
πs − 1−s
2
2
=
π
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
(−s)Γ
πs 2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ
Γ(s) = √ Γ
2
π
s
1
+
2 2
s
s
1−s
+ 1 ζ(1 − s) = π − 2 (s − 1)Γ
+ 1 ζ(s)
2
2
Function ξ(s)
ζ(1 − s) = cos
cos
π
πs − 1−s
2
2
πs π
=
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
(−s)Γ
2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ(s) = √ Γ
Γ
2
π
s
1
+
2 2
s
s
1−s
+ 1 ζ(1 − s) = π − 2 (s − 1)Γ
+ 1 ζ(s)
2
|
{z 2
}
ξ(s)
Function ξ(s)
ζ(1 − s) = cos
cos
π
|
πs − 1−s
2
2
πs π
=
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
(−s)Γ
2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ(s) = √ Γ
Γ
2
π
1
s
+
2 2
s
s
1−s
+ 1 ζ(1 − s) = π − 2 (s − 1)Γ
+ 1 ζ(s)
2
{z 2
}
{z
} |
ξ(1−s)
ξ(s)
Function ξ(s)
ζ(1 − s) = cos
cos
π
|
πs − 1−s
2
2
πs π
=
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
(−s)Γ
2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ(s) = √ Γ
Γ
2
π
1
s
+
2 2
s
s
1−s
+ 1 ζ(1 − s) = π − 2 (s − 1)Γ
+ 1 ζ(s)
2
{z 2
}
{z
} |
ξ(s)
ξ(1−s)
ξ(1 − s) = ξ(s)
Function ξ(s)
ζ(1 − s) = cos
cos
π
|
πs − 1−s
2
2
πs π
=
s
1
Γ(− 2 + 2 )Γ( 2s + 21 )
(−s)Γ
2
21−s π −s Γ(s)ζ(s)
2s−1 s Γ(s) = √ Γ
Γ
2
π
1
s
+
2 2
s
s
1−s
+ 1 ζ(1 − s) = π − 2 (s − 1)Γ
+ 1 ζ(s)
2
{z 2
}
{z
} |
ξ(s)
ξ(1−s)
ξ(1 − s) = ξ(s)
Function ξ(s) is entire, its zeroes are exactly non-trivial (i.e., non-real)
zeroes of ζ(s).
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
s
2
+ 1 ζ(s)
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
Ξ(t) = ξ
1
+ it
2
s
2
+ 1 ζ(s)
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
Ξ(t) = ξ
1
+ it
2
s
2
+ 1 ζ(s)
1
=ξ 1−
+ it
2
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
Ξ(t) = ξ
1
+ it
2
s
2
+ 1 ζ(s)
1
=ξ 1−
+ it
= Ξ(−t)
2
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
Ξ(t) = ξ
1
+ it
2
s
2
+ 1 ζ(s)
1
=ξ 1−
+ it
= Ξ(−t)
2
Ξ(t) is an even entire function
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
Ξ(t) = ξ
1
+ it
2
s
2
+ 1 ζ(s)
1
=ξ 1−
+ it
= Ξ(−t)
2
Ξ(t) is an even entire functionwith real value for real values of its
argument t.
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
Ξ(t) = ξ
1
+ it
2
s
2
+ 1 ζ(s)
1
=ξ 1−
+ it
= Ξ(−t)
2
Ξ(t) is an even entire functionwith real value for real values of its
argument t.
The Riemann Hypothesis. All zeroes of Ξ(t) are real numbers.
Riemann’s Hypothesis (RH)
s
ξ(s) = π − 2 (s − 1)Γ
Ξ(t) = ξ
1
+ it
2
s
2
+ 1 ζ(s)
1
=ξ 1−
+ it
= Ξ(−t)
2
Ξ(t) is an even entire functionwith real value for real values of its
argument t.
The Riemann Hypothesis. All zeroes of Ξ(t) are real numbers.
Chebyshev function ψ(x)
Chebyshev function ψ(x)
ψ(x) =
X
q≤x
q is a power
of prime p
ln(p)
Chebyshev function ψ(x)
ψ(x) =
X
ln(p)
q≤x
q is a power
of prime p
= ln(LCM(1, 2, ..., bxc))
Chebyshev function ψ(x)
ψ(x) =
X
ln(p)
q≤x
q is a power
of prime p
= ln(LCM(1, 2, ..., bxc))
20
15
10
5
0
5
10
15
20
Von Mangoldt Theorem
ψ(x) =
X
ln(p)
q≤x
q is a power
of prime p
Theorem (Hans Carl Fridrich von Mangoldt [1895]). For any
non-integer x > 1
ψ(x) = x −
X x ρ X x −2n
−
− ln(2π)
ρ
−2n
n
ξ(ρ)=0
Von Mangoldt Theorem
ψ(x) =
X
ln(p)
q≤x
q is a power
of prime p
Theorem (Hans Carl Fridrich von Mangoldt [1895]). For any
non-integer x > 1
ψ(x) = x −
X x ρ X x −2n
−
− ln(2π)
ρ
−2n
n
ξ(ρ)=0
Von Mangoldt Theorem
x−
X
ξ(ρ) = 0
|ρ| < 50
x ρ X x −2n
−
− ln(2π)
ρ
−2n
n
20
15
10
5
5
10
15
20
Von Mangoldt Theorem
x−
X
ξ(ρ) = 0
|ρ| < 100
x ρ X x −2n
−
− ln(2π)
ρ
−2n
n
20
15
10
5
5
10
15
20
Von Mangoldt Theorem
x−
X
ξ(ρ) = 0
|ρ| < 200
x ρ X x −2n
−
− ln(2π)
ρ
−2n
n
20
15
10
5
5
10
15
20
Von Mangoldt Theorem
x−
X
ξ(ρ) = 0
|ρ| < 400
x ρ X x −2n
−
− ln(2π)
ρ
−2n
n
20
15
10
5
5
10
15
20
Blowing the Trumpet
Assuming that all zeroes of Ξ(t) are real and simple, let them be
denoted ±γ1 , ±γ2 , . . . with 0 < γ1 < γ2 < . . . , thus the non-trivial
zeroes of ζ(s) are 12 ± iγ1 , 12 ± iγ2 , . . .
Blowing the Trumpet
Assuming that all zeroes of Ξ(t) are real and simple, let them be
denoted ±γ1 , ±γ2 , . . . with 0 < γ1 < γ2 < . . . , thus the non-trivial
zeroes of ζ(s) are 12 ± iγ1 , 12 ± iγ2 , . . .
-Γ6-Γ5 -Γ-Γ
3 2 -Γ1
Γ1
Γ2 Γ3
Γ5 Γ6 Γ7
Blowing the Trumpet
Assuming that all zeroes of Ξ(t) are real and simple, let them be
denoted ±γ1 , ±γ2 , . . . with 0 < γ1 < γ2 < . . . , thus the non-trivial
zeroes of ζ(s) are 12 ± iγ1 , 12 ± iγ2 , . . .
-Γ6-Γ5 -Γ-Γ
3 2 -Γ1
Γ1
Γ2 Γ3
Γ5 Γ6 Γ7
γ1 = 14.1347 . . .
γ2 = 21.0220 . . .
γ3 = 25.0109 . . .
γ4 = 30.4249 . . .
Blowing the Trumpet
Assuming that all zeroes of Ξ(t) are real and simple, let them be
denoted ±γ1 , ±γ2 , . . . with 0 < γ1 < γ2 < . . . , thus the non-trivial
zeroes of ζ(s) are 12 ± iγ1 , 12 ± iγ2 , . . .
-Γ6-Γ5 -Γ-Γ
3 2 -Γ1
Γ1
Γ2 Γ3
Γ5 Γ6 Γ7
γ1 = 14.1347 . . .
γ2 = 21.0220 . . .
γ3 = 25.0109 . . .
γ4 = 30.4249 . . .
Suppose that we have found γ1 , γ2 , . . . , γN−1 ; how could these
numbers be used for calculating an (approximate) value of the next
zero γN ?
Interpolating determinant
Let us try to approximate Ξ(t) by some simpler function also having
zeroes at the points ±γ1 , . . . , ±γN−1 .
Consider an interpolating determinant with even functions f1 , f2 , . . .
f1 (γ1 ) . . . f1 (γN−1 ) f1 (t) ..
..
.. ..
.
.
.
. fN (γ1 ) . . . fN (γN−1 ) fN (t)
Interpolating determinant
Let us try to approximate Ξ(t) by some simpler function also having
zeroes at the points ±γ1 , . . . , ±γN−1 .
Consider an interpolating determinant with even functions f1 , f2 , . . .
f1 (γ1 ) . . . f1 (γN−1 ) f1 (t) ..
..
.. ..
.
.
.
. fN (γ1 ) . . . fN (γN−1 ) fN (t)
Selecting fn (t) = t 2(n−1) we would obtain just an interpolating
polynomial
C
N−1
Y
n=1
having no other zeros.
(t 2 − γn 2 )
Interpolating determinant
Interpolating determinant
If γ1∗ , γ2∗ , . . . are zeros of the function
∗
Ξ (t) =
N
X
fk (t)
k=1
then the determinant
f1 (γ ∗ ) . . .
1
..
..
.
.
fN (γ ∗ ) . . .
1
vanish at every zero of Ξ∗ (t).
∗
f1 (γN−1
)
..
.
f1 (t) .. . ∗
fN (γN−1 ) fN (t)
Selecting f1 , f2 , . . .
Selecting f1 , f2 , . . .
Ξ(t) =
∞
X
n=1
1
αn (t)
αn (t) = −
it
π− 4 − 2 t 2 +
2n
1
4 Γ
1
+it
2
1
4
+
it
2
Selecting f1 , f2 , . . .
Ξ(t) =
∞
X
1
αn (t) = −
αn (t)
n=1
Ξ(t) =
∞
X
n=1
αn (t)
it
π− 4 − 2 t 2 +
2n
1
4 Γ
1
+it
2
1
4
+
it
2
Selecting f1 , f2 , . . .
Ξ(t) =
∞
X
1
αn (t) = −
αn (t)
n=1
Ξ(t) =
∞
X
n=1
αn (t)
it
π− 4 − 2 t 2 +
2n
Ξ(t) = Ξ(−t) =
1
4 Γ
1
+it
2
∞
X
n=1
1
4
+
αn (−t)
it
2
Selecting f1 , f2 , . . .
Ξ(t) =
∞
X
1
αn (t) = −
αn (t)
n=1
Ξ(t) =
∞
X
αn (t)
n=1
for =(t) < −
it
π− 4 − 2 t 2 +
2n
Ξ(t) = Ξ(−t) =
1
4 Γ
1
+it
2
∞
X
n=1
1
2
1
4
+
αn (−t)
it
2
Selecting f1 , f2 , . . .
Ξ(t) =
∞
X
1
αn (t) = −
αn (t)
n=1
Ξ(t) =
∞
X
αn (t)
n=1
for =(t) < −
it
π− 4 − 2 t 2 +
2n
Ξ(t) = Ξ(−t) =
1
4 Γ
1
+it
2
∞
X
n=1
1
2
for =(t) >
1
2
1
4
+
αn (−t)
it
2
Selecting f1 , f2 , . . .
Ξ(t) =
∞
X
1
αn (t) = −
αn (t)
n=1
Ξ(t) =
∞
X
αn (t)
2n
Ξ(t) = Ξ(−t) =
n=1
1
4 Γ
1
+it
2
∞
X
n=1
for =(t) < −
Ξ(t) =
it
π− 4 − 2 t 2 +
1
2
for =(t) >
∞
X
αn (t) + αn (−t)
n=1
2
1
2
1
4
+
αn (−t)
it
2
Selecting f1 , f2 , . . .
Ξ(t) =
∞
X
1
αn (t) = −
αn (t)
it
π− 4 − 2 t 2 +
2n
n=1
Ξ(t) =
∞
X
αn (t)
Ξ(t) = Ξ(−t) =
n=1
∞
X
n=1
for =(t) < −
Ξ(t) =
1
4 Γ
1
+it
2
1
2
for =(t) >
∞
X
αn (t) + αn (−t)
n=1
2
=
∞
X
n=1
1
2
βn (t)
1
4
+
αn (−t)
it
2
Our interpolating determinant
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
β1 (t) .. . βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
Our interpolating determinant
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
1
βn (t) =
it
π− 4 + 2 t 2 +
4n
1
4 Γ
1
−it
2
1
4
−
it
2
β1 (t) .. . βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
1
−
it
π− 4 − 2 t 2 +
4n
1
4 Γ
1
+it
2
1
4
+
it
2
Our interpolating determinant
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
1
βn (t) =
it
π− 4 + 2 t 2 +
4n
1
4 Γ
1
−it
2
1
4
Ξ(t) =
−
β1 (t) .. . βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
it
2
1
−
it
π− 4 − 2 t 2 +
4n
∞
X
n=1
βn (t)
1
4 Γ
1
+it
2
1
4
+
it
2
Blowing the trumpet: the outcome
Blowing the trumpet: the outcome
γ40 = 122.94682929355258. . .
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
γ220 = 427.20882508407458052814. . .
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
γ400 = 679.7421978825282177195259389112699953456135514. . .
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
∆400 ( 679.74219788252821771952593891126999534)=
−2.95319 . . . · 10−52001 < 0
γ400 = 679.7421978825282177195259389112699953456135514. . .
∆400 ( 679.74219788252821771952593891126999535)=
+1.78976 . . . · 10−52001 > 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
∆40 ( 124.25681)=+9.32826 . . . · 10−927 > 0
γ41 = 124.2568185543457. . .
∆40 ( 124.25682)=−2.20319 . . . · 10−925 < 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
∆40 ( 124.25681)=+9.32826 . . . · 10−927 > 0
γ41 = 124.2568185543457. . .
∆40 ( 124.25682)=−2.20319 . . . · 10−925 < 0
∆40 ( 127.516)=−2.24237 . . . · 10−924 < 0
γ42 = 127.51668387959. . .
∆40 ( 127.517)=+8.95138 . . . · 10−925 > 0
Blowing the trumpet: the outcome
∆40 ( 122.946829)=−1.86119 . . . · 10−926 < 0
γ40 = 122.94682929355258. . .
∆40 ( 122.946830)=+2.39694 . . . · 10−926 > 0
∆40 ( 124.25681)=+9.32826 . . . · 10−927 > 0
γ41 = 124.2568185543457. . .
∆40 ( 124.25682)=−2.20319 . . . · 10−925 < 0
∆40 ( 127.516)=−2.24237 . . . · 10−924 < 0
γ42 = 127.51668387959. . .
∆40 ( 127.517)=+8.95138 . . . · 10−925 > 0
∆40 ( 129.5787)=+2.78973 . . . · 10−926 > 0
γ43 = 129.578704199956. . .
∆40 ( 129.5788)=−9.94403 . . . · 10−927 < 0
Blowing the trumpet: the outcome
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
Blowing the trumpet: the outcome
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
∆220 ( 428.127914076616)=+3.30722 . . . · 10−17792 > 0
γ221 = 428.12791407661668211030. . .
∆220 ( 428.127914076617)=−1.28498 . . . · 10−17792 < 0
Blowing the trumpet: the outcome
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
∆220 ( 428.127914076616)=+3.30722 . . . · 10−17792 > 0
γ221 = 428.12791407661668211030. . .
∆220 ( 428.127914076617)=−1.28498 . . . · 10−17792 < 0
∆220 ( 430.3287454309386)=−1.08026 . . . · 10−17794 < 0
γ222 = 430.328745430938636669926. . .
∆220 ( 430.3287454309387)=+1.56602 . . . · 10−17793 > 0
Blowing the trumpet: the outcome
∆220 ( 427.208825084074)=−1.92776 . . . · 10−17793 < 0
γ220 = 427.20882508407458052814. . .
∆220 ( 427.208825084075)=+9.85564 . . . · 10−17794 > 0
∆220 ( 428.127914076616)=+3.30722 . . . · 10−17792 > 0
γ221 = 428.12791407661668211030. . .
∆220 ( 428.127914076617)=−1.28498 . . . · 10−17792 < 0
∆220 ( 430.3287454309386)=−1.08026 . . . · 10−17794 < 0
γ222 = 430.328745430938636669926. . .
∆220 ( 430.3287454309387)=+1.56602 . . . · 10−17793 > 0
......................................................
∆220 ( 441.683199201)=−3.85957 . . . · 10−17794 < 0
γ230 = 441.68319920118902387. . .
∆220 ( 441.683199202)=+1.39118 . . . · 10−17793 > 0
Blowing the trumpet: the outcome
∆12000 (t) has zeroes having more than 2000 common decimal digits
with γ12000 , γ12001 , . . . , γ12010
Partial explanation
Ξ(t) =
∞
X
n=1
βn (t)
Partial explanation
Ξ(t) =
∞
X
βn (t)
n=1
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
β1 (t) .. . βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
Partial explanation
Ξ(t) =
∞
X
βn (t)
n=1
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
β1 (t) N
.. = X δ̃ β (t)
N,n n
. n=1
βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
Partial explanation
Ξ(t) =
∞
X
βn (t)
n=1
β1 (γ1 ) . . . β1 (γN−1 ) β1 (t) N
X
..
.
.
.
.
.
.
∆N (t) = .
δ̃N,n βn (t)
.
.
. =
n=1
βN (γ1 ) . . . βN (γN−1 ) βN (t)
β1 (γ1 ) . . .
β1 (γN−1 ) ..
..
..
.
.
.
N+n βn−1 (γ1 ) . . . βn−1 (γN−1 ) δ̃N,n = (−1)
βn+1 (γ1 ) . . . βn+1 (γN−1 ) ..
..
..
.
.
.
βN (γ1 ) . . . βN (γN−1 ) Normalization
Ξ(t) =
∞
X
βn (t)
n=1
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
β1 (t) N
.. = X δ̃ β (t)
N,n n
. n=1
βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
Normalization
Ξ(t) =
∞
X
βn (t)
n=1
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
δN,n =
β1 (t) N
.. = X δ̃ β (t)
N,n n
. n=1
βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
δ̃N,n
δ̃N,1
Normalization
Ξ(t) =
∞
X
βn (t)
n=1
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
δN,n =
β1 (t) N
.. = X δ̃ β (t)
N,n n
. n=1
βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
δ̃N,n
δ̃N,1
δN,1 = 1
Normalization
Ξ(t) =
∞
X
βn (t)
n=1
β1 (γ1 ) . . .
..
∆N (t) = ...
.
βN (γ1 ) . . .
β1 (t) N
.. = X δ̃ β (t)
N,n n
. n=1
βN (γN−1 ) βN (t)
β1 (γN−1 )
..
.
δN,n =
δ̃N,n
δ̃N,1
N
X
n=1
δN,1 = 1
δN,n βn (t)
Normalized coefficients δ47,n
Normalized coefficients δ47,n
47
X
n=1
δ47,n βn (t)
Normalized coefficients δ47,n
Normalized coefficients δ47,n
δ47,n ≈ 1 + λ47 log(n)
Normalized coefficients δ47,n with logarithmic scale
Normalized coefficients δ47,n with logarithmic scale
δ47,n ≈ 1 + λ47 log(n)
Normalized coefficients δ48,n with logarithmic scale
Normalized coefficients δ48,n with logarithmic scale
δ48,n ≈ 1 + λ48 log(n)
Normalized coefficients δ49,n with logarithmic scale
Normalized coefficients δ49,n with logarithmic scale
δ49,n ≈ 1 + λ49 log(n)
Normalized coefficients δ50,n with logarithmic scale
Normalized coefficients δ50,n with logarithmic scale
δ50,n ≈ 1 + λ50 log(n)
Normalized coefficients δ51,n with logarithmic scale
Normalized coefficients δ51,n with logarithmic scale
δ51,n ≈ 1 + λ52 log(n)
Normalized coefficients δ52,n with logarithmic scale
Normalized coefficients δ52,n with logarithmic scale
δ52,n ≈ 1 + λ52 log(n)
Normalized coefficients δ53,n with logarithmic scale
Normalized coefficients δ53,n with logarithmic scale
δ53,n ≈ 1 + λ53 log(n)
Normalized coefficients δ54,n with logarithmic scale
Normalized coefficients δ54,n with logarithmic scale
δ54,n ≈ 1 + λ54 log(n)
Normalized coefficients δ55,n with logarithmic scale
Normalized coefficients δ55,n with logarithmic scale
δ55,n ≈ 1 + λ55 log(n)
Normalized coefficients δ56,n with logarithmic scale
Normalized coefficients δ56,n with logarithmic scale
δ56,n ≈ 1 + λ56 log(n)
Normalized coefficients δ57,n with logarithmic scale
Normalized coefficients δ57,n with logarithmic scale
δ57,n ≈ 1 + λ57 log(n)
Normalized coefficients δ130,n with logarithmic scale
Normalized coefficients δ130,n with logarithmic scale
Normalized coefficients δ130,n with logarithmic scale
δ130,n ≈ 1 + µ130,2 dom2 (n) + λ130 log(n)
The characteristic function of the divisibility
domm (k) =
1, if m | k
0, otherwise
Normalized coefficients δ130,n with logarithmic scale
δ130,n ≈ 1 + µ130,2 dom2 (n) + λ130 log(n)
Normalized coefficients δ131,n with logarithmic scale
Normalized coefficients δ131,n with logarithmic scale
δ131,n ≈ 1 + µ131,2 dom2 (n) + λ131 log(n)
Normalized coefficients δ132,n with logarithmic scale
Normalized coefficients δ132,n with logarithmic scale
δ132,n ≈ 1 + µ132,2 dom2 (n) + λ132 log(n)
Normalized coefficients δ169,n with logarithmic scale
Normalized coefficients δ169,n with logarithmic scale
δ169,n ≈ 1 + µ169,2 dom2 (n) + λ169 log(n)
Normalized coefficients δ180,n with logarithmic scale
Normalized coefficients δ180,n with logarithmic scale
δ180,n ≈ 1 + µ180,2 dom2 (n) + λ180 log(n)
Normalized coefficients δ220,n
Normalized coefficients δ220,n
δ220,n ≈ 1 + µ220,2 dom2 (n) + λ220 log(n)
Normalized coefficients δ500,n
Normalized coefficients δ500,n
Normalized coefficients δ1200,n
Normalized coefficients δ1200,n
Normalized coefficients δ3000,n
Normalized coefficients δ3000,n
Partial explanation
The function ∆220 (t) is a “smooth” truncation, not of the divergent
Diriclet series
∞
∞
X
X
αn (t) + αn (−t)
Ξ(t) =
βn (t) =
2
n=1
n=1
Partial explanation
The function ∆220 (t) is a “smooth” truncation, not of the divergent
Diriclet series
∞
∞
X
X
αn (t) + αn (−t)
Ξ(t) =
βn (t) =
2
n=1
n=1
but of the convergent (for real t) alternating series
∞
∞
X
X
αn (t) + αn (−t)
(−1)n−1 βn (t) =
(−1)n−1
2
n=1
n=1
Partial explanation
The function ∆220 (t) is a “smooth” truncation, not of the divergent
Diriclet series
∞
∞
X
X
αn (t) + αn (−t)
Ξ(t) =
βn (t) =
2
n=1
n=1
but of the convergent (for real t) alternating series
∞
∞
X
X
αn (t) + αn (−t)
(−1)n−1 βn (t) =
(−1)n−1
2
n=1
n=1
!
1
it
1
it
∞
X
π − 4 + 2 Γ 41 − it2
π − 4 − 2 Γ 41 + it2
n−1
2
1
+
=
(−1)
t +4
1
1
−it
2
4n
4n 2 +it
n=1
∞
1
1 2 1 − 1 + it 1 it X
=
t + 4 π 4 2Γ 4 − 2
(−1)n−1 n− 2 +it +
4
n=1
1 2
t +
4
1
4
1
it
π− 4 − 2 Γ
1
4
+
∞
X
it
2
n=1
1
(−1)n−1 n− 2 −it
Normalized coefficients δ321,n
A Conjecture
A Conjecture. For every real ν such that 1 ≤ ν the sequence
δbνc,1 , . . . , (−1)n−1 δbνnc,n , . . .
has certain limiting value δ(ν).
Normalized coefficients δ505,n
Normalized coefficients δ621,n
Normalized coefficients δ810,n
Normalized coefficients δ5600,n
Normalized coefficients δ8000,n
Normalized coefficients δ12000,n
Normalized coefficients δ11981,n
Normalized coefficients δ321,n
Normalized coefficients δ321,n
Normalized coefficients δ321,n
Normalized coefficients δ321,n
δ321,n ≈ 1 + µ321,2 dom2 (n) + µ321,3 dom3 (n) + λ321 log(n)
Normalized coefficients δ321,n
δ321,n ≈ 1 + µ321,2 dom2 (n) + µ321,3 dom3 (n) + λ321 log(n)
µ321,2 = −2 − 4.98 . . . · 10−13
µ321,3 = 7.47 . . . · 10−13
λ321 = −3.33 . . . · 10−18
Normalized coefficients: general case
Normalized coefficients: general case
δN,n ≈
P
m
µN,m domm (n) + λN log(n)
Normalized coefficients: general case
δN,n ≈
P
m
µN,m domm (n) + λN log(n)
µN,1 = 1
dom1 (m) ≡ 1
Averaged normalized coefficients
δN,n,a =
δN,n + · · · + δN,n+a−1
a
Averaged normalized coefficients
δN,n,a =
δN,n,2 =
δN,n + · · · + δN,n+a−1
a
δN,n + δN,n+1
2
Averaged normalized coefficients
δN,n,a =
δN,n + · · · + δN,n+a−1
a
δN,n + δN,n+1
2
dom2 (n) + dom2 (n + 1)
≈ 1 + µN,2
+
2
dom3 (n) + dom3 (n + 1)
log(n(n + 1))
µN,3
+ λN
2
2
δN,n,2 =
Averaged normalized coefficients
δN,n,a =
δN,n + · · · + δN,n+a−1
a
δN,n + δN,n+1
2
dom2 (n) + dom2 (n + 1)
≈ 1 + µN,2
+
2
dom3 (n) + dom3 (n + 1)
log(n(n + 1))
µN,3
+ λN
2
2
µN,2
= 1+
+
2
dom3 (n) + dom3 (n + 1)
log(n(n + 1))
µN,3
+ λN
2
2
δN,n,2 =
Averaged normalized coefficients δ321,n,2
µ321,3 = 7.47 . . . · 10−13
Averaged normalized coefficients δ999,n,2
Averaged normalized coefficients δ999,n,2
µ999,3 = 2.26 . . . · 10−41
Averaged normalized coefficients δ999,n,6
µ999,4 = 1.75 . . . · 10−90
Averaged normalized coefficients δ999,n,12
µ999,5 = −3.09 . . . · 10−127
Averaged normalized coefficients δ999,n,60
Averaged normalized coefficients δ999,n,60
Averaged normalized coefficients δ999,n,60
λ999 = −1.73 . . . · 10−144
Differences δ999,n,60 − λ999 log(Γ(n + 60) − Γ(n))/60
λ999 = −1.73 . . . · 10−144
µ999,7 = 1.52 . . . · 10−171
Almost linear relations
Almost linear relations
r1 δN,1 + r2 δN,2 + · · · + rm δN,m = r
r1 , r2 , . . . , rm , and r are either rational numbers with small
denominators or very close to integers
νN,n =
n
Pn−1
k=1
νN,n
µN,k
k
µN,1 = 1
µN,n
N = 3200
νN,n /µN,n
P
νN,n = n−1
k=1
n
νN,n
2
1
µN,k
k
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
P
νN,n = n−1
k=1
n
νN,n
2
1
µN,k
k
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1
=−2.00... · 100
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
−7.37... · 10304
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1
=−2.00... · 100
µN,2
= 1 + 5.42... · 10−305
µN,4
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
−7.37... · 10304
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1
=−1.99... · 100
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
−6.73... · 10547
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1
=−1.99... · 100
µN,2
= 1 + 2.25... · 10−134
µN,6
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
−6.73... · 10547
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2
=+4.50... · 10−134
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
+1.51... · 10414
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2
=+4.50... · 10−134
µN,3
= 1 − 3.95... · 10−414
µN,6
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
+1.51... · 10414
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1
=−2.00... · 100
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
−2.16... · 10691
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1
=−2.00... · 100
µN,2
= 1 + 5.42... · 10−305
µN,8
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
−2.16... · 10691
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1 − µN,2
=+1.08... · 10−304
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
+1.17... · 10387
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1 − µN,2
=+1.08... · 10−304
µN,4
= 1 − 6.82... · 10−387
µN,8
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
+1.17... · 10387
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1 − µN,2 −
µN,4 =+7.40... · 10−691
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
8 − 2.52... · 10−46
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=−2.00... · 100
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1 − µN,2 −
µN,4 =+7.40... · 10−691
N = 3200
νN,n /µN,n
2 + 3.00... · 10−134
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
8 − 2.52... · 10−46
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
9
−2.91... · 10−738
µN,1 = δN,1 = 1
µN,n
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1 − µN,2 −
µN,4 =+7.40... · 10−691
δN,9 − µN,1
=+4.50... · 10−134
N = 3200
νN,n /µN,n
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
8 − 2.52... · 10−46
+1.54... · 10604
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
9
−2.91... · 10−738
µN,1 = δN,1 = 1
µN,n
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1 − µN,2 −
µN,4 =+7.40... · 10−691
δN,9 − µN,1
=+4.50... · 10−134
µN,3
= 1 − 5.63... · 10−604
µN,9
N = 3200
νN,n /µN,n
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
8 − 2.52... · 10−46
+1.54... · 10604
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
3
−1.50... · 10−134
4
−2.71... · 10−305
5
−4.72... · 10−443
6
−2.97... · 10−548
7
+5.96... · 10−630
8
−9.25... · 10−692
9
−2.91... · 10−738
µN,1 = δN,1 = 1
µN,n
δN,3 − µN,1
=+4.50... · 10−134
δN,4 − µN,1 − µN,2
=+1.08... · 10−304
δN,5 − µN,1
=+2.36... · 10−442
δN,6 − µN,1 − µN,2 −
µN,3 =+1.78... · 10−547
δN,7 − µN,1
=−4.17... · 10−629
δN,8 − µN,1 − µN,2 −
µN,4 =+7.40... · 10−691
δN,9 − µN,1 − µN,3
=+2.54... · 10−737
N = 3200
νN,n /µN,n
3 − 5.41... · 10−171
4 − 6.96... · 10−138
5 − 3.14... · 10−105
6 + 1.20... · 10−81
7 + 1.08... · 10−61
8 − 2.52... · 10−46
+8.71... · 100
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
3
−4.70... · 10−201
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
µN,1 = δN,1 = 1
µN,n
δN,3 − µN,1
=+1.41... · 10−200
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
N = 4800
νN,n /µN,n
3 + 1.63... · 10−259
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
3
−4.70... · 10−201
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
µN,1 = δN,1 = 1
µN,n
δN,3 − µN,1
=+1.41... · 10−200
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
N = 4800
νN,n /µN,n
3 + 1.63... · 10−259
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
10
+1.88... · 10−1213
µN,1 = δN,1 = 1
µN,n
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
δN,10 − µN,1
=−2.00... · 100
N = 4800
νN,n /µN,n
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
+1.05... · 101213
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
10
+1.88... · 10−1213
µN,1 = δN,1 = 1
µN,n
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
δN,10 − µN,1
=−2.00... · 100
µN,2
= 1 + 3.64... · 10−671
µN,10
N = 4800
νN,n /µN,n
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
+1.05... · 101213
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
10
+1.88... · 10−1213
µN,1 = δN,1 = 1
µN,n
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
δN,10 − µN,1 − µN,2
=+7.29... · 10−671
N = 4800
νN,n /µN,n
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
−3.86... · 10542
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
10
+1.88... · 10−1213
µN,1 = δN,1 = 1
µN,n
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
δN,10 − µN,1 − µN,2
=+7.29... · 10−671
µN,5
= 1 + 2.58... · 10−542
µN,10
N = 4800
νN,n /µN,n
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
−3.86... · 10542
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
10
+1.88... · 10−1213
µN,1 = δN,1 = 1
µN,n
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
δN,10 − µN,1 − µN,2 −
µN,5 =−1.88... · 10−1212
N = 4800
νN,n /µN,n
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
10 + 2.81... · 10−35
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
4
+2.56... · 10−460
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
10
+1.88... · 10−1213
µN,1 = δN,1 = 1
µN,n
δN,4 − µN,1 − µN,2
=−1.02... · 10−459
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
δN,10 − µN,1 − µN,2 −
µN,5 =−1.88... · 10−1212
N = 4800
νN,n /µN,n
4 + 2.27... · 10−211
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
10 + 2.81... · 10−35
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
5
−1.45... · 10−671
6
−1.33... · 10−836
7
+9.49... · 10−966
8
−1.06... · 10−1067
9
+6.38... · 10−1149
10
+1.88... · 10−1213
11
−5.32... · 10−1249
µN,1 = δN,1 = 1
µN,n
δN,5 − µN,1
=+7.29... · 10−671
δN,6 − µN,1 − µN,2 −
µN,3 =+8.01... · 10−836
δN,7 − µN,1
=−6.64... · 10−965
δN,8 − µN,1 − µN,2 −
µN,4 =+8.49... · 10−1067
δN,9 − µN,1 − µN,3
=−5.74... · 10−1148
δN,10 − µN,1 − µN,2 −
µN,5 =−1.88... · 10−1212
δN,11 − µN,1
=+2.13... · 10−1249
N = 4800
νN,n /µN,n
5 − 4.58... · 10−165
6 + 4.26... · 10−129
7 + 7.82... · 10−102
8 + 4.81... · 10−81
9 − 2.66... · 10−64
10 + 2.81... · 10−35
+4.00... · 10−1
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
5
−9.62... · 10−517
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
µN,1 = δN,1 = 1
µN,n
δN,5 − µN,1
=+4.81... · 10−516
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
N = 8000
νN,n /µN,n
5 − 1.07... · 10−287
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
5
−9.62... · 10−517
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
µN,1 = δN,1 = 1
µN,n
δN,5 − µN,1
=+4.81... · 10−516
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
N = 8000
νN,n /µN,n
5 − 1.07... · 10−287
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1
=+1.98... · 10285
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+6.41... · 101954
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1
=+1.98... · 10285
µN,2
= 1 − 3.00... · 100
µN,12
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+6.41... · 101954
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2
=+5.96... · 10285
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+1.92... · 101955
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2
=+5.96... · 10285
µN,3
= 1 − 2.61... · 10−439
µN,12
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+1.92... · 101955
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2 −
µN,3 =+1.56... · 10−153
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+5.03... · 101516
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2 −
µN,3 =+1.56... · 10−153
µN,4
= 1 − 7.94... · 10−651
µN,12
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+5.03... · 101516
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4
=+1.24... · 10−803
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+4.00... · 10866
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4
=+1.24... · 10−803
µN,6
= 1 − 2.99... · 10−866
µN,12
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
+4.00... · 10866
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+3.71... · 10−1669
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
12 + 2.07... · 10−67
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
6
−2.06... · 10−804
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
µN,1 = δN,1 = 1
µN,n
δN,6 − µN,1 − µN,2 −
µN,3 =+1.24... · 10−803
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+3.71... · 10−1669
N = 8000
νN,n /µN,n
6 − 2.99... · 10−229
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
12 + 2.07... · 10−67
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
7
−1.03... · 10−1033
8
−3.45... · 10−1218
9
−9.12... · 10−1367
10
−2.01... · 10−1488
11
−3.37... · 10−1589
12
−3.09... · 10−1670
13
+5.36... · 10−1738
µN,1 = δN,1 = 1
µN,n
δN,7 − µN,1
=+7.22... · 10−1033
δN,8 − µN,1 − µN,2 −
µN,4 =+2.76... · 10−1217
δN,9 − µN,1 − µN,3
=+8.21... · 10−1366
δN,10 − µN,1 − µN,2 −
µN,5 =+2.01... · 10−1487
δN,11 − µN,1
=+3.70... · 10−1588
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+3.71... · 10−1669
δN,13 − µN,1
=−6.97... · 10−1737
N = 8000
νN,n /µN,n
7 − 2.34... · 10−184
8 − 2.11... · 10−148
9 − 1.98... · 10−121
10 − 1.67... · 10−100
11 − 1.01... · 10−80
12 + 2.07... · 10−67
13 − 2.13... · 10−4
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
7
−2.24... · 10−779
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
µN,1 = δN,1 = 1
µN,n
δN,7 − µN,1
=+1.57... · 10−778
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
N = 9600
νN,n /µN,n
7 − 5.54... · 10−224
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
7
−2.24... · 10−779
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
µN,1 = δN,1 = 1
µN,n
δN,7 − µN,1
=+1.57... · 10−778
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
N = 9600
νN,n /µN,n
7 − 5.54... · 10−224
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
µN,1 = δN,1 = 1
µN,n
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1
=−3.52... · 10818
N = 9600
νN,n /µN,n
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
+1.06... · 102543
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
µN,1 = δN,1 = 1
µN,n
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1
=−3.52... · 10818
µN,2
= 1 + 4.45... · 10−1597
µN,14
N = 9600
νN,n /µN,n
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
+1.06... · 102543
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
µN,1 = δN,1 = 1
µN,n
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2
=+1.57... · 10−778
N = 9600
νN,n /µN,n
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
−4.74... · 10946
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
µN,1 = δN,1 = 1
µN,n
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2
=+1.57... · 10−778
µN,7
= 1 + 2.95... · 10−946
µN,14
N = 9600
νN,n /µN,n
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
−4.74... · 10946
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
µN,1 = δN,1 = 1
µN,n
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2 −
µN,7 =−4.64... · 10−1724
N = 9600
νN,n /µN,n
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
14 − 1.59... · 10−6
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
8
−1.78... · 10−1003
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
µN,1 = δN,1 = 1
µN,n
δN,8 − µN,1 − µN,2 −
µN,4 =+1.42... · 10−1002
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2 −
µN,7 =−4.64... · 10−1724
N = 9600
νN,n /µN,n
8 − 1.41... · 10−183
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
14 − 1.59... · 10−6
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
15
+3.78... · 10−1732
µN,1 = δN,1 = 1
µN,n
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2 −
µN,7 =−4.64... · 10−1724
δN,15 − µN,1
=+5.29... · 10818
N = 9600
νN,n /µN,n
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
14 − 1.59... · 10−6
−1.39... · 102550
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
15
+3.78... · 10−1732
µN,1 = δN,1 = 1
µN,n
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2 −
µN,7 =−4.64... · 10−1724
δN,15 − µN,1
=+5.29... · 10818
µN,3
= 1 + 3.43... · 10−968
µN,15
N = 9600
νN,n /µN,n
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
14 − 1.59... · 10−6
−1.39... · 102550
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
15
+3.78... · 10−1732
µN,1 = δN,1 = 1
µN,n
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2 −
µN,7 =−4.64... · 10−1724
δN,15 − µN,1 − µN,3
=−1.81... · 10−149
N = 9600
νN,n /µN,n
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
14 − 1.59... · 10−6
+4.80... · 101582
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
15
+3.78... · 10−1732
µN,1 = δN,1 = 1
µN,n
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2 −
µN,7 =−4.64... · 10−1724
δN,15 − µN,1 − µN,3
=−1.81... · 10−149
µN,5
= 1 − 1.14... · 10−1633
µN,15
N = 9600
νN,n /µN,n
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
14 − 1.59... · 10−6
+4.80... · 101582
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
9
−3.14... · 10−1187
10
−9.07... · 10−1339
11
+1.66... · 10−1464
12
−4.46... · 10−1568
13
+1.35... · 10−1654
14
+3.31... · 10−1725
15
+3.78... · 10−1732
µN,1 = δN,1 = 1
µN,n
δN,9 − µN,1 − µN,3
=+2.83... · 10−1186
δN,10 − µN,1 − µN,2 −
µN,5 =+9.07... · 10−1338
δN,11 − µN,1
=−1.83... · 10−1463
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+5.35... · 10−1567
δN,13 − µN,1
=−1.75... · 10−1653
δN,14 − µN,1 − µN,2 −
µN,7 =−4.64... · 10−1724
δN,15 − µN,1 − µN,3 −
µN,5 =−2.08... · 10−1782
N = 9600
νN,n /µN,n
9 − 2.59... · 10−151
10 + 1.84... · 10−125
11 + 2.94... · 10−103
12 + 3.63... · 10−86
13 − 3.19... · 10−70
14 − 1.59... · 10−6
+5.52... · 10−51
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
9
−5.68... · 10−1326
10
+9.29... · 10−1525
11
+2.27... · 10−1686
12
−9.06... · 10−1825
13
−3.75... · 10−1939
14
+3.29... · 10−2036
15
+9.24... · 10−2118
µN,1 = δN,1 = 1
µN,n
δN,9 − µN,1 − µN,3
=+5.11... · 10−1325
δN,10 − µN,1 − µN,2 −
µN,5 =−9.29... · 10−1524
δN,11 − µN,1
=−2.50... · 10−1685
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+1.08... · 10−1823
δN,13 − µN,1
=+4.88... · 10−1938
δN,14 − µN,1 − µN,2 −
µN,7 =−4.61... · 10−2035
δN,15 − µN,1 − µN,3 −
µN,5 =−1.38... · 10−2116
N = 12000
νN,n /µN,n
9 + 1.47... · 10−198
10 − 2.44... · 10−161
11 + 4.37... · 10−138
12 − 4.96... · 10−114
13 + 1.14... · 10−96
14 − 3.93... · 10−81
15 − 7.70... · 10−27
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
9
−5.68... · 10−1326
10
+9.29... · 10−1525
11
+2.27... · 10−1686
12
−9.06... · 10−1825
13
−3.75... · 10−1939
14
+3.29... · 10−2036
15
+9.24... · 10−2118
µN,1 = δN,1 = 1
µN,n
δN,9 − µN,1 − µN,3
=+5.11... · 10−1325
δN,10 − µN,1 − µN,2 −
µN,5 =−9.29... · 10−1524
δN,11 − µN,1
=−2.50... · 10−1685
δN,12 − µN,1 − µN,2 −
µN,3 − µN,4 − µN,6
=+1.08... · 10−1823
δN,13 − µN,1
=+4.88... · 10−1938
δN,14 − µN,1 − µN,2 −
µN,7 =−4.61... · 10−2035
δN,15 − µN,1 − µN,3 −
µN,5 =−1.38... · 10−2116
N = 12000
νN,n /µN,n
9 + 1.47... · 10−198
10 − 2.44... · 10−161
11 + 4.37... · 10−138
12 − 4.96... · 10−114
13 + 1.14... · 10−96
14 − 3.93... · 10−81
15 − 7.70... · 10−27
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
+3.10... · 10828
4
−1.65... · 10552
5
+1.99... · 10−1
6
−8.36... · 10−445
7
+1.82... · 10−799
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=+6.21... · 10828
δN,3 − µN,1
=−9.32... · 10828
δN,4 − µN,1 − µN,2
=+6.62... · 10552
δN,5 − µN,1
=−1.00... · 100
δN,6 − µN,1 − µN,2 −
µN,3 =+5.01... · 10−444
δN,7 − µN,1
=−1.27... · 10−798
N = 12000
νN,n /µN,n
−6.21... · 10828
3 + 1.59... · 10−276
4 + 4.83... · 10−553
5 + 2.09... · 10−443
6 + 1.31... · 10−354
7 + 1.09... · 10−288
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
+3.10... · 10828
4
−1.65... · 10552
5
+1.99... · 10−1
6
−8.36... · 10−445
7
+1.82... · 10−799
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=+6.21... · 10828
δN,3 − µN,1
=−9.32... · 10828
δN,4 − µN,1 − µN,2
=+6.62... · 10552
δN,5 − µN,1
=−1.00... · 100
δN,6 − µN,1 − µN,2 −
µN,3 =+5.01... · 10−444
δN,7 − µN,1
=−1.27... · 10−798
N = 12000
νN,n /µN,n
−6.21... · 10828
3 + 1.59... · 10−276
4 + 4.83... · 10−553
5 + 2.09... · 10−443
6 + 1.31... · 10−354
7 + 1.09... · 10−288
δ12000,2 = 6.21856447151825627740964946899... · 10828
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
+3.10... · 10828
4
−1.65... · 10552
5
+1.99... · 10−1
6
−8.36... · 10−445
7
+1.82... · 10−799
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=+6.21... · 10828
δN,3 − µN,1
=−9.32... · 10828
δN,4 − µN,1 − µN,2
=+6.62... · 10552
δN,5 − µN,1
=−1.00... · 100
δN,6 − µN,1 − µN,2 −
µN,3 =+5.01... · 10−444
δN,7 − µN,1
=−1.27... · 10−798
N = 12000
νN,n /µN,n
−6.21... · 10828
3 + 1.59... · 10−276
4 + 4.83... · 10−553
5 + 2.09... · 10−443
6 + 1.31... · 10−354
7 + 1.09... · 10−288
δ12000,2 = 6.21856447151825627740964946899... · 10828
δ12000,3 = −9.32784670727738441611447420348... · 10828
νN,n =
n
Pn−1
k=1
µN,k
k
νN,n
2
1
3
+3.10... · 10828
4
−1.65... · 10552
5
+1.99... · 10−1
6
−8.36... · 10−445
7
+1.82... · 10−799
µN,1 = δN,1 = 1
µN,n
δN,2 − µN,1
=+6.21... · 10828
δN,3 − µN,1
=−9.32... · 10828
δN,4 − µN,1 − µN,2
=+6.62... · 10552
δN,5 − µN,1
=−1.00... · 100
δN,6 − µN,1 − µN,2 −
µN,3 =+5.01... · 10−444
δN,7 − µN,1
=−1.27... · 10−798
N = 12000
νN,n /µN,n
−6.21... · 10828
3 + 1.59... · 10−276
4 + 4.83... · 10−553
5 + 2.09... · 10−443
6 + 1.31... · 10−354
7 + 1.09... · 10−288
δ12000,2 = 6.21856447151825627740964946899... · 10828
δ12000,3 = −9.32784670727738441611447420348... · 10828
δ12000,5 = −2.38085755689986864713671438596 · 10−548
log |δ12000,n |
log |δ12000,n |