International Mathematical Forum, 3, 2008, no. 28, 1383 - 1386
Sums of Prime Numbers, the Zeta Function
and the π Number
Rafael Jakimczuk
Division Matematica, Universidad Nacional de Lujan
Buenos Aires, Argentina
jakimczu@mail.unlu.edu.ar
Abstract
Let Sm (n) be the sum of the m-th powers of the prime factors in the
prime factorization of n!.
We prove the following asymptotic formula
Sm (n) ∼
ζ(m + 1) nm+1
m + 1 log n
Where ζ(s) is the Riemann’s Zeta Function.
Let σm,p (n) be the sum of the m-th powers of the different prime
divisors of n.
We prove the following asymptotic formula
n
i=1
σm,p (i) ∼
ζ(m + 1) nm+1
m + 1 log n
Mathematics Subject Classification: 11A41
Keywords: Sums of prime numbers, the zeta function, the π number
1
Lemmas
The following theorems are well known [1]. We shall use them as lemmas.
Let π(x) be the number of primes not exceeding x.
Lemma 1.1 (Prime number theorem). The following formula holds
x
x
+o
π(x) =
log x
log x
1384
R. Jakimczuk
Lemma 1.2 If p ≤ n is a prime, then
n
n
n
+ 2 + 3 + ...
p
p
p
is the exponent of p appearing in the prime factorization of n!.
Let Ω(n) be the number of prime factors in the prime factorization of n.
Lemma 1.3 The following formula holds
Ω(n!) =
n
Ω(i) = n log log n + o (n log log n)
i=1
The following lemma is a consequence of the prime number theorem ( see [3]
or [2]).
Let sm (x) be the sum of the m-th powers of the primes not exceeding x,
where m = 1, 2, 3, . . .
Lemma 1.4
xm+1
xm+1
+o
p =
sm (x) =
(m + 1) log x
log x
p≤x
m
Finally, we can obtain without difficulty the following lemma.
Lemma 1.5
∞
j=1
1
j
−
m
j
(j + 1)m+1
= ζ(m + 1)
(m = 1, 2, 3, . . .)
where ζ(s) is the Riemann’s Zeta Function.
Remark. Note that in the proof of lemma 1.5 ( if m = 1) there exists the
formula
1
1
1
n
+
+ ...+
=
1.2 2.3
n(n + 1)
n+1
2
Main Results
Let Sm (n) be the sum of the m-th powers of the prime factors in the prime
factorization of n!.
Theorem 2.1 The following asymptotic formula holds
Sm (n) ∼
ζ(m + 1) nm+1
m + 1 log n
(1)
1385
Sums of prime numbers and the zeta function
Proof. Lemma 1.2 gives
m
p≤n
√
√
p≤ n
p
p
√
p
n
p
m
n<p≤n
√
Note that
p
n
<p≤ n
3
2
n
<p≤ n
j+1
j
n
j+1
1
p =
m+1
<p≤ n
m
j
n
<p≤ n
j+1
j
pm
p
√
n<p≤n
p
n
p
j m+1
=
√
n
n<p≤ k+1
log log n + o n
n
<p≤ n
k+1
k
n
=j
p
m
m+2
2
p
n
+
p
m
log log n
n
n
<p≤ k−1
k
(3)
p
m
n
p
(4)
2
1
1
n
=
p
m+1
m
n
n
+
pm
p
p
n
<p≤n
pm
n
<p≤ n
j+1
j
1
−
(j + 1)m+1
j
1
−
m
j
(j + 1)m+1
Substituting (7) into (4) we obtain
n
n
+ 2 + ...
p
p
Substituting (6) into (5) we find that
n
n
+ 2 + ...
p
p
Lemma 1.4 gives
= n
m+2
2
n
+
p
m
n
n<p≤ k+1
+ ...+
≤ n
=
m
2
p≤n
Now, we have
(2)
n
n
+ 2 + ...
p
p
m
n
p
m
n<p≤n
Lemma 1.3 gives
+
m
n
n
+ 2 + ... =
p
√
p
p
p≤ n
p
Sm (n) =
p
pm
(5)
nm+1
nm+1
+o
log n
log n
nm+1
nm+1
+o
log n
log n
(6)
(7)
m
n
p
k
nm+1 1 j
1
+
−
m
log n m + 1 j=1 j
(j + 1)m+1
nm+1
+o
log n
(8)
Equations (2), (3), (8) and lemma 1.5 imply (1). The theorem is thus proved.
Lemma 1.4 and theorem 2.1 imply the following relation between the prime
numbers and the Riemann’s Zeta Function.
1386
R. Jakimczuk
Corollary 2.2 The following limit holds
lim
n→∞
Sm (n)
= ζ(m + 1)
sm (n)
In particular, if m is odd, we obtain infinite relations between the prime numbers and the π number from a well known Euler’s theorem. For example:
S1 (n)
π2
= ζ(2) =
n→∞ s1 (n)
6
S3 (n)
π4
= ζ(4) =
n→∞ s3 (n)
90
lim
lim
Let σm,p (n) be the sum of the m-th powers of the different prime divisors of n.
Theorem 2.3 The following asymptotic formula holds
n
i=1
Proof. We have
σm,p (i) ∼
n
i=1
ζ(m + 1) nm+1
m + 1 log n
σm,p (i) =
p
p≤n
m
n
p
Now, the proof is the same as in theorem 2.1. Theorem 2.3 is proved.
Corollary 2.4 The following limit holds
n
lim
n→∞
i=1 σm,p (i)
sm (n)
= ζ(m + 1)
References
[1] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers,
Oxford, 1960.
[2] R. Jakimczuk, A note on sums of powers which have a fixed number of
prime factors , Journal of Inequalities in Pure and Applied Mathematics,
Volume 6, Issue 2, (2005), Article 31.
[3] T. Salat and S. Znam, On the sums of prime powers, Acta Fac. Rer. Nat.
Univ. Com. Math., 21 (1968), 21 - 25.
Received: January 24, 2008