G APS BETWEEN ZEROS OF L- FUNCTIONS
Caroline Turnage-Butterbaugh
North Dakota State University / Duke University
Elementary, analytic, and algorithmic number theory:
Research inspired by the mathematics of Carl Pomerance
University of Georgia
June 9, 2015
T HE R IEMANN ZETA - FUNCTION
Let s denote a complex number.
∞
X
Y 1
1 −1
ζ(s) :=
,
=
1− s
ns
p
n=1
<(s) > 1
p prime
1 / 28
T HE R IEMANN ZETA - FUNCTION
Let s denote a complex number.
∞
X
Y 1
1 −1
ζ(s) :=
,
=
1− s
ns
p
n=1
<(s) > 1
p prime
Via the Euler product, knowledge about the nontrivial zeros of
ζ(s) reveals knowledge about the distribution of the primes.
1 / 28
T HE R IEMANN ZETA - FUNCTION
Let s denote a complex number.
∞
X
Y 1
1 −1
ζ(s) :=
,
=
1− s
ns
p
n=1
<(s) > 1
p prime
Via the Euler product, knowledge about the nontrivial zeros of
ζ(s) reveals knowledge about the distribution of the primes.
Riemann Hypothesis (RH)
The nontrivial zeros of ζ(s) have real part equal to 1/2.
1 / 28
Let ρ = β + iγ denote a nontrivial zero of ζ(s). Then we have
N(T) :=
X
ρ
0<γ<T
1∼
T
log T.
2π
2 / 28
Let ρ = β + iγ denote a nontrivial zero of ζ(s). Then we have
N(T) :=
X
ρ
0<γ<T
1∼
T
log T.
2π
Consider the sequence of ordinates of zeros in the critical strip:
0 < γ1 ≤ γ2 ≤ . . . ≤ γn ≤ γn+1 ≤ . . .
2 / 28
Let ρ = β + iγ denote a nontrivial zero of ζ(s). Then we have
N(T) :=
X
1∼
ρ
0<γ<T
T
log T.
2π
Consider the sequence of ordinates of zeros in the critical strip:
0 < γ1 ≤ γ2 ≤ . . . ≤ γn ≤ γn+1 ≤ . . .
The average size of γn+1 − γn is
2π
.
log(γn )
2 / 28
Let
µ := lim inf
n→∞
γn+1 − γn
(2π/ log γn )
and
λ := lim sup
n→∞
γn+1 − γn
.
(2π/ log γn )
3 / 28
Let
µ := lim inf
n→∞
γn+1 − γn
(2π/ log γn )
and
λ := lim sup
n→∞
γn+1 − γn
.
(2π/ log γn )
By definition, we have
µ ≤ 1 ≤ λ,
3 / 28
Let
µ := lim inf
n→∞
γn+1 − γn
(2π/ log γn )
and
λ := lim sup
n→∞
γn+1 − γn
.
(2π/ log γn )
By definition, we have
µ ≤ 1 ≤ λ,
and we expect
µ=0
and
λ = ∞.
3 / 28
Let
µ := lim inf
n→∞
γn+1 − γn
(2π/ log γn )
and
λ := lim sup
n→∞
γn+1 − γn
.
(2π/ log γn )
By definition, we have
µ ≤ 1 ≤ λ,
and we expect
µ=0
and
λ = ∞.
Conjecture
Gaps between consecutive zeros of ζ(s) that are arbitrarily small
(large), relative to the average gap size, appear infinitely often.
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
R.R. Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui
3.18
Preobrazhenskii
0.515396
a∗ Results are unconditional, but RH must be assumed to give lower
bound on λ. a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
R.R. Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui
3.18
Preobrazhenskii
0.515396
a∗ Results are unconditional, but RH must be assumed to give lower
bound on λ. a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
R.R. Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui
3.18
Preobrazhenskii
0.515396
a∗ Results are unconditional, but RH must be assumed to give lower
bound on λ. a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui
3.18
Preobrazhenskii
0.515396
a∗ Results are unconditional; must assume RH for lower bound on λ.
a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui
3.18
Preobrazhenskii
0.515396
a∗ Results are unconditional; must assume RH for lower bound on λ.
a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui
3.18
Preobrazhenskii
0.515396
a∗ Results are unconditional; must assume RH for lower bound on λ.
a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui
3.18
Preobrazhenskii
0.515396
a∗ Results are unconditional; must assume RH for lower bound on λ.
a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui∗∗
3.18
Preobrazhenskii∗∗
0.515396
a∗ Results are unconditional; must assume RH for lower bound on λ.
a∗∗ preprint
3 / 28
W HAT IS KNOWN ? ( ASSUMING RH)
lower bound for λ upper bound for µ
Gonek/Mueller
1.9
Montgomery & Odlyzko
1.9799
0.5179
Conrey, Ghosh & Gonek
2.3378
0.5172
Hall
2.6306∗
Bui, Milinovich & Ng
2.6950
0.5155
Feng & Wu
2.7327
0.5154
Bredberg
2.766∗
Bui∗∗
3.18
Preobrazhenskii∗∗
0.515396
a∗ Results are unconditional; must assume RH for lower bound on λ.
a∗∗ preprint
3 / 28
G APS BETWEEN ZEROS OF OTHER L- FUNCTIONS
• For large gaps, we consider the following degree 2
L-functions:
• ζK (s) – the Dedekind zeta-function of a quadratic number
field K
4 / 28
G APS BETWEEN ZEROS OF OTHER L- FUNCTIONS
• For large gaps, we consider the following degree 2
L-functions:
• ζK (s) – the Dedekind zeta-function of a quadratic number
field K
• L(s, f ) – an automorphic L-function on GL(2) over Q, where
f is either a primitive holomorphic cusp form or a primitive
Maass cusp form
4 / 28
G APS BETWEEN ZEROS OF OTHER L- FUNCTIONS
• For large gaps, we consider the following degree 2
L-functions:
• ζK (s) – the Dedekind zeta-function of a quadratic number
field K
• L(s, f ) – an automorphic L-function on GL(2) over Q, where
f is either a primitive holomorphic cusp form or a primitive
Maass cusp form
• For small gaps, we consider primitive L-functions from the
Selberg Class.
4 / 28
D EDEKIND ZETA - FUNCTION OF A NUMBER FIELD
Let K be a number field.
The Dedekind zeta-function attached to K is defined by
ζK (s) =
X
I⊂OK
−1
Y 1
1
1−
,
=
N(I)s
N(p)s
<(s) > 1.
p⊂OK
5 / 28
D EDEKIND ZETA - FUNCTION OF A NUMBER FIELD
Let K be a number field.
The Dedekind zeta-function attached to K is defined by
ζK (s) =
X
I⊂OK
−1
Y 1
1
1−
,
=
N(I)s
N(p)s
<(s) > 1.
p⊂OK
√
Let K = Q[ d], where d is a fundamental discriminant (positive
or negative).
In this case,
ζK (s) = ζ(s)L(s, χd ).
5 / 28
D EDEKIND ZETA - FUNCTION OF A NUMBER FIELD
Let K be a number field.
The Dedekind zeta-function attached to K is defined by
ζK (s) =
X
I⊂OK
−1
Y 1
1
1−
,
=
N(I)s
N(p)s
<(s) > 1.
p⊂OK
√
Let K = Q[ d], where d is a fundamental discriminant (positive
or negative).
In this case,
ζK (s) = ζ(s)L(s, χd ).
Note that ζK (s) is imprimitive.
5 / 28
A UTOMORPHIC L- FUNCTIONS ON GL(2) OVER Q
Let f denote a primitive holomorphic cusp form on GL(2) over
Q with level q, which we consider to be fixed.
Then f has a Fourier expansion of the form
f (z) =
∞
X
af (n)n(k−1)/2 e(nz),
=(z) > 0.
n=1
6 / 28
A UTOMORPHIC L- FUNCTIONS ON GL(2) OVER Q
Let f denote a primitive holomorphic cusp form on GL(2) over
Q with level q, which we consider to be fixed.
Then f has a Fourier expansion of the form
f (z) =
∞
X
af (n)n(k−1)/2 e(nz),
=(z) > 0.
n=1
From this, the associated L-function is defined as
L(s, f ) =
∞
X
af (n)
n=1
ns
Y
αf (p) −1
βf (p) −1
=
1−
1−
,
ps
ps
p
<(s) 1
6 / 28
A UTOMORPHIC L- FUNCTIONS ON GL(2) OVER Q
Let f denote a primitive holomorphic cusp form on GL(2) over
Q with level q, which we consider to be fixed.
Then f has a Fourier expansion of the form
f (z) =
∞
X
af (n)n(k−1)/2 e(nz),
=(z) > 0.
n=1
From this, the associated L-function is defined as
L(s, f ) =
∞
X
af (n)
n=1
ns
Y
αf (p) −1
βf (p) −1
=
1−
1−
,
ps
ps
p
<(s) 1
Note that L(s, f ) is primitive.
6 / 28
Z EROS OF ζK (s)
Let ρK = βK + iγK denote a nontrivial zero of ζK (s). We have
p
X
|d|T
TLd
NK (T) :=
1∼
,
Ld := log
.
2
π
4π
ρ
0<γ<T
7 / 28
Z EROS OF ζK (s)
Let ρK = βK + iγK denote a nontrivial zero of ζK (s). We have
p
X
|d|T
TLd
NK (T) :=
1∼
,
Ld := log
.
2
π
4π
ρ
0<γ<T
Consider the sequence of ordinates of zeros in the critical strip:
0 < γK (1) ≤ γK (2) ≤ . . . ≤ γK (n) ≤ γK (n + 1) ≤ . . .
7 / 28
Z EROS OF ζK (s)
Let ρK = βK + iγK denote a nontrivial zero of ζK (s). We have
p
X
|d|T
TLd
NK (T) :=
1∼
,
Ld := log
.
2
π
4π
ρ
0<γ<T
Consider the sequence of ordinates of zeros in the critical strip:
0 < γK (1) ≤ γK (2) ≤ . . . ≤ γK (n) ≤ γK (n + 1) ≤ . . .
The average size of γK (n + 1) − γK (n) is
π
p
.
log( |d|γK (n))
7 / 28
Z EROS OF L(s, f )
Let ρf = βf + iγf denote a nontrivial zero of L(s, f ). We have
p
X
TLq
|q|T
Nf (T) :=
1∼
,
Lq := log
.
π
4π 2
ρ
0<γf <T
Consider the sequence of ordinates of zeros in the critical strip:
0 < γf (1) ≤ γf (2) ≤ . . . ≤ γf (n) ≤ γf (n + 1) ≤ . . .
The average size of γf (n + 1) − γf (n) is
π
p
.
log( |q|γf (n))
8 / 28
As in the case of ζ(s), we expect that
λK := lim sup
γK (n + 1) − γK (n)
p
=∞
π/(log( |d|γK (n)))
λf := lim sup
γf (n + 1) − γf (n)
p
= ∞.
π/(log( |q|γf (n)))
n→∞
and
n→∞
9 / 28
As in the case of ζ(s), we expect that
λK := lim sup
γK (n + 1) − γK (n)
p
=∞
π/(log( |d|γK (n)))
λf := lim sup
γf (n + 1) − γf (n)
p
= ∞.
π/(log( |q|γf (n)))
n→∞
and
n→∞
Conjecture
Gaps between consecutive zeros that are arbitrarily large, relative to
the average gap size, appear infinitely often for both ζK (s) and L(s, f ).
9 / 28
L ARGE GAPS BETWEEN ZEROS OF ζK (s)
Theorem (T., 2014)
Assuming GRH for ζK (s), we have λK ≥ 2.449.
10 / 28
L ARGE GAPS BETWEEN ZEROS OF ζK (s)
Theorem (T., 2014)
Assuming GRH for ζK (s), we have λK ≥ 2.449.
Theorem (Bui, Heap, T., 2014 (preprint))
Assuming GRH for ζK (s), we have λK ≥ 2.866.
Bui
Heap
T.
Both of these results can be stated unconditionally if we restrict our
attention to zeros on the critical line.
10 / 28
L ARGE GAPS BETWEEN ZEROS OF L(s, f )
Theorem (Barrett, McDonald, Miller, Ryan, T., Winsor, 2015)
Assuming GRH for L(s, f ), we have λf ≥ 1.732.
Barrett, McDonald, Miller, Ryan, T., Winsor
(Research conducted at SMALL 2014 at Williams College.)
This result can also be stated unconditionally if we restrict our
attention to zeros on the critical line.
11 / 28
H ALL’ S M ETHOD ( MODIFIED BY B REDBERG )
Wirtinger’s Inequality
Let g : [a, b] → C be continuously differentiable and suppose that
g(a) = g(b) = 0. Then
Z
b
2
|g(t)| dt ≤
a
b−a
π
2 Z
b
|g0 (t)|2 dt.
a
12 / 28
H ALL’ S M ETHOD ( MODIFIED BY B REDBERG )
Wirtinger’s Inequality
Let g : [a, b] → C be continuously differentiable and suppose that
g(a) = g(b) = 0. Then
Z
b
2
|g(t)| dt ≤
a
b−a
π
2 Z
b
|g0 (t)|2 dt.
a
By understanding the mean-values of g(t) and g0 (t), we can
obtain a lower bound on gaps between zeros of g(t).
12 / 28
H ALL’ S M ETHOD ( MODIFIED BY B REDBERG )
Wirtinger’s Inequality
Let g : [a, b] → C be continuously differentiable and suppose that
g(a) = g(b) = 0. Then
Z
b
2
|g(t)| dt ≤
a
b−a
π
2 Z
b
|g0 (t)|2 dt.
a
By understanding the mean-values of g(t) and g0 (t), we can
obtain a lower bound on gaps between zeros of g(t).
This reveals the necessity to restrict ourselves to degree 2
L-functions.
12 / 28
M EAN - VALUES FOR THE CASE L(1/2+it, f )
Theorem (Barrett, McDonald, Miller, Ryan, T., Winsor, 2015)
Let µ, ν be non-negative integers. Then
Z
2T
T
L(µ) ( 12 +it, f )L(ν) ( 12 +it, f ) dt
∼
(−1)µ+ν 2µ+ν+1
cf T(log T)µ+ν+1
µ+ν+1
as T → ∞, where cf denotes the residue of the simple pole of
L(s, f × f ) at s = 1.
13 / 28
M EAN - VALUES FOR THE CASE L(1/2+it, f )
Theorem (Barrett, McDonald, Miller, Ryan, T., Winsor, 2015)
Let µ, ν be non-negative integers. Then
Z
2T
T
L(µ) ( 12 +it, f )L(ν) ( 12 +it, f ) dt
∼
(−1)µ+ν 2µ+ν+1
cf T(log T)µ+ν+1
µ+ν+1
as T → ∞, where cf denotes the residue of the simple pole of
L(s, f × f ) at s = 1.
Good, 1975: µ = ν = 0
Yashiro, 2014: µ = ν
(f a holomorphic cusp form of even weight with respect to SL2 (Z))
13 / 28
I DEA OF ARGUMENT FOR ζK (s)
Let
g(t) := exp(iνLt)ζK ( 12 + it)M( 12 + it),
where ν is a real constant that will be chosen later, and M(s) is
an amplifier of the form
14 / 28
I DEA OF ARGUMENT FOR ζK (s)
Let
g(t) := exp(iνLt)ζK ( 12 + it)M( 12 + it),
where ν is a real constant that will be chosen later, and M(s) is
an amplifier of the form
M(s) =
X dr (h1 )dr (h2 )χd (h2 )P[h1 h2 ]
(h1 h2 )s
h1 h2 ≤y
where y = Tθ , 0 < θ < 1/4, and dr (h) denotes the coefficients of
ζ(s)r .
14 / 28
I DEA OF ARGUMENT FOR ζK (s)
Let
g(t) := exp(iνLt)ζK ( 12 + it)M( 12 + it),
where ν is a real constant that will be chosen later, and M(s) is
an amplifier of the form
M(s) =
X dr (h1 )dr (h2 )χd (h2 )P[h1 h2 ]
(h1 h2 )s
h1 h2 ≤y
where y = Tθ , 0 < θ < 1/4, and dr (h) denotes the coefficients of
ζ(s)r .
Here
P[h] = P
log y/h
log y
for 1 ≤ h ≤ y and P(x) is a polynomial.
14 / 28
I DEA OF ARGUMENT FOR ζK (s)
λK := lim sup
n→∞
γK (n + 1) − γK (n)
p
(π/ log |d|γK (n))
15 / 28
I DEA OF ARGUMENT FOR ζK (s)
λK := lim sup
n→∞
γK (n + 1) − γK (n)
p
(π/ log |d|γK (n))
Assume (towards contradiction) that
λK ≤ κ.
15 / 28
I DEA OF ARGUMENT FOR ζK (s)
λK := lim sup
n→∞
γK (n + 1) − γK (n)
p
(π/ log |d|γK (n))
Assume (towards contradiction) that
λK ≤ κ.
Let
t1 ≤ t2 ≤ . . . ≤ tN
denote the zeros of g(t) in the interval [T, 2T].
15 / 28
I DEA OF ARGUMENT FOR ζK (s)
λK := lim sup
n→∞
γK (n + 1) − γK (n)
p
(π/ log |d|γK (n))
Assume (towards contradiction) that
λK ≤ κ.
Let
t1 ≤ t2 ≤ . . . ≤ tN
denote the zeros of g(t) in the interval [T, 2T].
By our assumption,
tn+1 − tn ≤ (1 + o(1))
κπ
.
Ld
15 / 28
By Wirtinger’s Inequality,
Z
tn+1
tn
|g(t)|2 dt ≤ (1 + o(1))
κ2
Ld 2
Z
tn+1
|g0 (t)|2 dt.
tn
16 / 28
By Wirtinger’s Inequality,
Z
tn+1
|g(t)|2 dt ≤ (1 + o(1))
tn
κ2
Ld 2
Z
tn+1
|g0 (t)|2 dt.
tn
Summing for zeros between height T and 2T, we have
Z
2T
T
κ2
|g(t)| dt ≤ (1 + o(1)) 2
Ld
2
Z
2T
|g0 (t)|2 dt.
T
16 / 28
By Wirtinger’s Inequality,
Z
tn+1
|g(t)|2 dt ≤ (1 + o(1))
tn
κ2
Ld 2
Z
tn+1
|g0 (t)|2 dt.
tn
Summing for zeros between height T and 2T, we have
Z
2T
T
Therefore, if
κ2
|g(t)| dt ≤ (1 + o(1)) 2
Ld
2
Z
2T
|g0 (t)|2 dt.
T
R 2T
Ld 2 T |g(t)|2 dt
> 1,
R
κ2 2T |g0 (t)|2 dt
T
we may conclude that
λK > κ.
16 / 28
M EAN - VALUES FOR THE CASE ζK (1/2+it)
Using a special case of a result of Bettin, Bui, Li, and Radziwiłł
(which computes the twisted moment of the product of four
Dirichlet L-functions) we have
17 / 28
M EAN - VALUES FOR THE CASE ζK (1/2+it)
Using a special case of a result of Bettin, Bui, Li, and Radziwiłł
(which computes the twisted moment of the product of four
Dirichlet L-functions) we have
Theorem (Bui, Heap, T., 2014)
We have
Z 2T
|g(t)|2 dt ∼
Cr (0)L2d
2 +4r+1
T + O(TL2r
)
d
2
4
(2r − 1)!((r − 1)!)
|g0 (t)|2 dt ∼
Cr (1)L4d
2 +4r+3
T + O(TL2r
)
d
2
4
(2r − 1)!((r − 1)!)
T
and
Z
2T
T
as T → ∞, where Cr (0), Cr (1) are constants depending on the
coefficients of ζK (s)r and are given explicitly.
17 / 28
λK := lim sup
n→∞
γK (n + 1) − γK (n)
p
(π/ log |d|γK (n))
18 / 28
λK := lim sup
n→∞
γK (n + 1) − γK (n)
p
(π/ log |d|γK (n))
The choice of θ = 1/4, ν = 1.2773, r = 1, and
P(x) = 1 − 10.8998x + 28.9444x2 − 22.1343x3 + 0.6148x4
allows us to conclude that
18 / 28
λK := lim sup
n→∞
γK (n + 1) − γK (n)
p
(π/ log |d|γK (n))
The choice of θ = 1/4, ν = 1.2773, r = 1, and
P(x) = 1 − 10.8998x + 28.9444x2 − 22.1343x3 + 0.6148x4
allows us to conclude that
Theorem (Bui, Heap, T., 2014)
We have λK ≥ 2.866. That is, there are infinitely many pairs of
consecutive zeros of ζK (s) that are more than 2.866 times the average
spacing apart.
18 / 28
R EMARK - L ENGTH OF THE AMPLIFIER
M(s) =
X dr (h1 )dr (h2 )χd (h2 )P[h1 h2 ]
,
(h1 h2 )s
θ
0 < θ < 1/4
h1 h2 ≤T
19 / 28
R EMARK - L ENGTH OF THE AMPLIFIER
M(s) =
X dr (h1 )dr (h2 )χd (h2 )P[h1 h2 ]
,
(h1 h2 )s
θ
0 < θ < 1/4
h1 h2 ≤T
Lower bounds for λK in terms of θ
in the simple case r = 1, P(x) = 1.
19 / 28
R EMARK - L ENGTH OF THE AMPLIFIER
M(s) =
X dr (h1 )dr (h2 )χd (h2 )P[h1 h2 ]
,
(h1 h2 )s
θ
0 < θ < 1/4
h1 h2 ≤T
• The function is not strictly
increasing as θ → 1,
(which is widely believed
to be the largest value in
the range of validity for
twisted moment results).
Lower bounds for λK in terms of θ
in the simple case r = 1, P(x) = 1.
19 / 28
R EMARK - L ENGTH OF THE AMPLIFIER
M(s) =
X dr (h1 )dr (h2 )χd (h2 )P[h1 h2 ]
,
(h1 h2 )s
θ
0 < θ < 1/4
h1 h2 ≤T
• The function is not strictly
increasing as θ → 1,
(which is widely believed
to be the largest value in
the range of validity for
twisted moment results).
• This phenomenon has also
Lower bounds for λK in terms of θ
in the simple case r = 1, P(x) = 1.
been observed by Bredberg
and Bui when studying
large gaps between zeros
of ζ(s).
19 / 28
R EMARK - HIGHER MOMENTS ?
It is not clear that emulating higher moments should
necessarily lead to larger gaps in our framework.
20 / 28
R EMARK - HIGHER MOMENTS ?
It is not clear that emulating higher moments should
necessarily lead to larger gaps in our framework.
The basic point is that the coefficients in the denominator of the
ratio
R 2T
T |g(t)|dt
R 2T
0
T |g (t)|dt
can often be larger than that of the numerator when one
considers higher moments.
20 / 28
R EMARK - HIGHER MOMENTS ?
It is not clear that emulating higher moments should
necessarily lead to larger gaps in our framework.
The basic point is that the coefficients in the denominator of the
ratio
R 2T
T |g(t)|dt
R 2T
0
T |g (t)|dt
can often be larger than that of the numerator when one
considers higher moments.
This is the subject of Hall’s (2002) JNT article.
20 / 28
S MALL GAPS BETWEEN ZEROS OF L- FUNCTIONS
• For small gaps, we take a different approach and employ a
pair correlation argument.
21 / 28
S MALL GAPS BETWEEN ZEROS OF L- FUNCTIONS
• For small gaps, we take a different approach and employ a
pair correlation argument.
• We are not restricted to degree 2 L-functions.
21 / 28
S MALL GAPS BETWEEN ZEROS OF L- FUNCTIONS
• For small gaps, we take a different approach and employ a
pair correlation argument.
• We are not restricted to degree 2 L-functions.
• We consider primitive L-functions from the Selberg Class.
21 / 28
T HE S ELBERG C LASS S
All L-functions L(s) ∈ S enjoy the following properties.
• Dirichlet Series:
L(s) =
∞
X
aL (n)
n=1
• Ramanujan Hypothesis:
ns
,
<(s) > 1
aL (n) no(1)
22 / 28
T HE S ELBERG C LASS S
All L-functions L(s) ∈ S enjoy the following properties.
• Dirichlet Series:
L(s) =
∞
X
aL (n)
n=1
• Ramanujan Hypothesis:
ns
,
<(s) > 1
aL (n) no(1)
• Analytic Continuation: There exists an integer a ≥ 0, such
that (s − 1)a L(s) is an entire function of finite order.
22 / 28
T HE S ELBERG C LASS S
All L-functions L(s) ∈ S enjoy the following properties.
• Dirichlet Series:
L(s) =
∞
X
aL (n)
n=1
• Ramanujan Hypothesis:
ns
,
<(s) > 1
aL (n) no(1)
• Analytic Continuation: There exists an integer a ≥ 0, such
that (s − 1)a L(s) is an entire function of finite order.
• Functional Equation: relating L(s) ←→ L(1 − s).
The degree, mL , of L(s) is encoded in the functional
equation.
22 / 28
T HE S ELBERG C LASS S
• Euler Product:
log L(s) =
∞
X
bL (n)
n=1
ns
,
where bL (n) = 0 unless n = p` for some ` ≥ 1 and
bL (n) nθ
for some θ < 1/2.
23 / 28
T HE S ELBERG C LASS S
• Euler Product:
log L(s) =
∞
X
bL (n)
n=1
ns
,
where bL (n) = 0 unless n = p` for some ` ≥ 1 and
bL (n) nθ
for some θ < 1/2.
Primitive members of the Selberg Class S include
• ζ(s), the Riemann zeta-function (degree m = 1)
• L(s, f ), where f is a primitive holomorphic cusp form
(degree m = 2)
23 / 28
S MALL GAPS - SET UP
Fix any L ∈ S. As before, consider the sequence
0 ≤ γL (1) ≤ γL (2) ≤ . . . ≤ γL (n) ≤ . . .
of consecutive ordinates of nontrivial zeros of L(s).
24 / 28
S MALL GAPS - SET UP
Fix any L ∈ S. As before, consider the sequence
0 ≤ γL (1) ≤ γL (2) ≤ . . . ≤ γL (n) ≤ . . .
of consecutive ordinates of nontrivial zeros of L(s).
Let
µL := lim inf
n→∞
γL (n + 1) − γL (n)
.
(average spacing)
As in the special case of small gaps between zeros of ζ(s), we
expect µL = 0.
24 / 28
Theorem (Barett, McDonald, Miller, Ryan, T., Winsor, 2015)
Let L ∈ S be primitive of degree mL . Assume GRH and Hypothesis
A. Then there is a computable nontrivial upper bound on µL
depending on mL . In particular,
mL
1
2
3
4
5
..
.
upper bound for µL
0.606894
0.822897
0.905604
0.942914
0.962190
..
.
where the nontrivial upper bounds for µL approach 1 as mL increases.
The case mL = 1 has previously been shown by Carneiro, Chandee,
Littmann, and Milinovich in 2014.
25 / 28
M ETHODS O VERVIEW
• We generalize an argument of Goldston, Gonek, Özlük,
and Snyder with a new modification of Carneiro, Chandee,
Littmann, and Milinovich.
26 / 28
M ETHODS O VERVIEW
• We generalize an argument of Goldston, Gonek, Özlük,
and Snyder with a new modification of Carneiro, Chandee,
Littmann, and Milinovich.
• Murty and Perelli proved a general pair correlation result
for all primitive L-functions in the Selberg Class for
restricted support inversely proportional to the degree of
the function, assuming the Generalized Riemann
Hypothesis and their Hypothesis A.
26 / 28
M ETHODS O VERVIEW
• We generalize an argument of Goldston, Gonek, Özlük,
and Snyder with a new modification of Carneiro, Chandee,
Littmann, and Milinovich.
• Murty and Perelli proved a general pair correlation result
for all primitive L-functions in the Selberg Class for
restricted support inversely proportional to the degree of
the function, assuming the Generalized Riemann
Hypothesis and their Hypothesis A.
• Hypothesis A is a mild assumption concerning the
correlation of the coefficients of L-functions at primes and
prime powers.
26 / 28
C URRENT W ORK : SMALL 2015
The large gaps results for L(s, f ), where f is either a primitive
holomorphic or Maass cusp form, can likely be improved.
27 / 28
C URRENT W ORK : SMALL 2015
The large gaps results for L(s, f ), where f is either a primitive
holomorphic or Maass cusp form, can likely be improved.
• We will use the argument employed by Bui, Heap, and T.
(for ζK (s)) to hopefully improve the large gaps result for
L(s, f ).
27 / 28
C URRENT W ORK : SMALL 2015
The large gaps results for L(s, f ), where f is either a primitive
holomorphic or Maass cusp form, can likely be improved.
• We will use the argument employed by Bui, Heap, and T.
(for ζK (s)) to hopefully improve the large gaps result for
L(s, f ).
• Note: this method has also lent itself to the current best
lower bound on λ for large gaps between zeros ζ(s) (Bui,
2014).
27 / 28
Thank you!
28 / 28