Summatory relations and prime products for the Stieltjes constants, and other related results

0.17 MB·

by  Mark W. Coffey

Summatory relations and prime products for the
Stieltjes constants, and other related results
7
1
0
2
Mark W. Coffey
n
a Department of Physics
J
Colorado School of Mines
9
1
Golden, CO 80401

] (Received 2016)
T
N
. July 31, 2016
h
t
a
m
Abstract
[

1 The Stieltjes constants γ (a) appear in the regular part of the Laurent ex-
k
v
pansion for the Hurwitz zeta function ζ(s,a). We present summatory results
4
6 for these constants γ (a) in terms of fundamental mathematical constants such
k
0
as the Catalan constant, and further relate them to products of rational func-
7
tions of prime numbers. We provide examples of infinite series of differences of
0
. Stieltjes constants evaluating as volumes in hyperbolic 3-space. We present a
1
0 new series representation for the difference of the first Stieltjes constant at ra-
7 tional arguments. We obtain expressions for ζ(1/2)L (1/2), where for primes
p
1 −
p > 7, L (s) are certain L-series, and remarkably tight bounds for the value
: p
v −
ζ(1/2), ζ(s)= ζ(s,1) being the Riemann zeta function.
i
X
r
a Key words and phrases
Dirichlet L function, Hurwitz zeta function, Stieltjes constants, series represen-
tation, Riemann zeta function, Laurent expansion, digamma function, summatory
relation, prime product
2010 AMS codes
11M06, 11Y60, 11M35
1
Introduction
Let ζ(s,a) be the Hurwitz zeta function, and ζ(s,1) = ζ(s) be the Riemann zeta
function [18, 23, 27, 30] . In the complex plane of s, each of these functions has a
simple pole of residue 1 at s = 1. For Re s > 1 and Re a > 0 we have
∞ 1
ζ(s,a) = , (1.1)
(n+a)s
n=0
X
and by analytic continuation ζ(s,a) extends to a meromorphic function through out
the whole complex plane. Accordingly, there is the Laurent expansion in terms of the
Stieltjes constants γ (a), [6, 9, 11, 12, 29, 31]
k
1 ∞ ( 1)n
ζ(s,a) = + − γ (a)(s 1)n, s = 1. (1.2)
n
s 1 n! − 6
− n=0
X
WeletΓ(s)betheGammafunction, ψ(s) = Γ(s)/Γ(s)bethedigammafunction(e.g.,
′
[1, 2, 20]) with the Euler constant γ = γ (1) = ψ(1), and recall that γ (a) = ψ(a).
0 0
− −
In the following, ψ(k) are the polygamma functions. We note that for integers n > 1,
ζ(n,a) reduces to values ψ(n 1).
−
The sequence γ (a) exhibits complicated changes in sign with k. For in-
{ k }∞k=0
stance, for both even and odd index, there are infinitely many positive and negative
values. Furthermore, there is sign variation with the parameter a. These features, as
well as the exponential growth in magnitude in k, are now well captured in an asymp-
totic expression ([24], Section 2, [26]). In fact, though initially derived for large values
of k, this expression is useful for computational approximation even for small values
of k.
2
Reciprocity and other relations for the Stieltjes constants, Bernoulli polynomials,
and functions A (q) are given in [10]. The works [13] and [12] present explicit expres-
k
sions for the initial values of the Stieltjes constants at rational argument, and various
series representations, respectively.
This paper is largely illustrative, being a step in a program to explore summatory
relations for the Stieltjes constants in terms of fundamental mathematical constants
such as the Catalan constant
∞ ( 1)n
G − . (1.3)
≡ (2n+1)2
n=0
X
We also relate sums of differences of Stieltjes constants to other mathematical con-
stants and products of prime numbers. 1. For the latter purpose, p is reserved for the
product index over primes. Two of many examples to be presented are then
∞ 1 1 2 1
γ γ = , (1.4a)
k k
k! 3 − 3 3
k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
and
π2 p2 1 1 ∞ 1 1 3
G = − = γ γ . (1.4b)
8 p2 +1 16 k! k 4 − k 4
p≡3 k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Y X
mod 4
While, for instance, the first equality in (1.4b) is known, it seems unlikely that the
connection with the Stieljtes constants in the second equality has been established
before.
Amajor toolforthefirst sections ofthispaper arequadraticDirichlet-Lseries and
their properties. Therefore, some further notation and definitions are introduced. Let
1
For such prime product results, especially see Proposition 7
3
D be a fundamental discriminant and (D/n) the Kronecker-Jacobi-Legendre symbol.
This symbol, a completely multiplicative function on the positive integers, is a real
primitive Dirichlet character withmodulus D . ThentheDirichlet Lseries associated
| |
to (D/n) is defined for Re s > 1 as
∞ D
L (s) = n s. (1.5)
D −
n
n=1(cid:18) (cid:19)
X
If D = 1, then L (s) = ζ(s). For all other values of D, L (s) can be made into an
1 D
entire function by using the value L (1) = 0. With suitable factors of Ds/2, π s/2,
D −
6
and Γ[(s + ǫ)/2] (ǫ = 0,1), L (s) may be completed to L (s) with the compact
D ∗D
functional equation L (s) = L (1 s). A useful compilation of special values of
∗D ∗D −
L (s) is contained in [25].
D
Importantly for what follows, we describe how Dirichlet L-functions L (s) (e.g.,
k
±
[22], Ch. 16), are expressible as linear combinations of Hurwitz zeta functions. We
let χ be a real Dirichlet character modulo k, where the corresponding L function is
k
written with subscript k according to χ (k 1) = 1. We have
k
± − ±
k
∞ χ (n) 1 m
k
L (s) = = χ (m)ζ s, , Re s > 1. (1.6)
±k ns ks k k
Xn=1 mX=1 (cid:16) (cid:17)
This equation holds for at least Re s > 1. If χ is a nonprincipal character, as we
k
typically assume in the following, then convergence obtains for Re s > 0.
These L functions, extendable to the whole complex plane, satisfy the functional
equations [33]
1 sπ
L (s) = (2π)sk s+1/2cos Γ(1 s)L (1 s), (1.7)
k − k
− π 2 − − −
(cid:16) (cid:17)
4
and
1 sπ
L (s) = (2π)sk s+1/2sin Γ(1 s)L (1 s). (1.8)
+k − +k
π 2 − −
(cid:16) (cid:17)
Owing to the relation
π
Γ(1 s)Γ(s) = , (1.9)
− sinπs
these functional equations may also be written in the form
πs
L (1 s) = 2(2π) sks 1/2sin Γ(s)L (s), (1.10)
k − − k
− − 2 −
(cid:16) (cid:17)
and
πs
L (1 s) = 2(2π) sks 1/2cos Γ(s)L (s). (1.11)
+k − − +k
− 2
(cid:16) (cid:17)
Statement of results
The first result may be placed in the context of special function theory, while the
subsequent ones Propositions 2–8 and 10 provide summatory results for differences of
the Stieltjes constants. Proposition 9 concerns representation of Sierpinski’s constant
S, and the last Proposition 11 bounds ζ(s) in the critical strip for real values of s.
Proposition 1 (a)
k
1 m
L (1) = χ (m)γ ,
k k 0
± k k
mX=1 (cid:16) (cid:17)
and (b) for all D = 1, the values
6
π if D = 3
3√3 −
π if D = 4
L (1) = 4 − ,
D πh(D) if D < 4
√ D −
2h−(D)lnε if D > 1
√D





5
may be written in terms of γ = ψ. Here h(D) is the ideal class number of the
0
−
quadratic field Q(√D) and ε is the fundamental unit of the integer subring Z+((D+
√D)/2)Z .
Proposition 2 (a)
1 ∞ ( 1)n k m
L (2) = − χ (m)γ ,
±k k2 n! k n k
Xn=0 mX=1 (cid:16) (cid:17)
and (b) for all D, the values L (2) may similarly be written in terms of sums of
D
differences γ (a). This includes thecases L (2) = ζ(2) = π2/6, L (2) = G, Catalan’s
k 1 4
−
constant,
4π2 π2 π2
L (2) = , L (2) = , L (2) = ,
5 8 12
25√5 8√2 6√3
and
24 π/2 tant+√7
L (2) = I ln dt. (2.1)
7 7
− − ≡ 7√7 Zπ/3 (cid:12)(cid:12)tant−√7(cid:12)(cid:12)
(cid:12) (cid:12)
The value (2.1) and related integrals arise in(cid:12)hyperbolic(cid:12)geometry, knot theory,
(cid:12) (cid:12)
and quantum field theory, and (2.1) has received considerable attention in the last
several years [3, 4, 5, 7]. References [14, 15] provide alternative evaluations of the
integral (2.1) in terms of the Clausen function Cl (θ),
2
θ t 1 xsinθ dx
Cl (θ) ln 2sin dt = tan 1
2 −
≡ − 2 1 xcosθ x
Z0 (cid:12) (cid:12) Z0 (cid:18) − (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
1(cid:12) (cid:12)lnx ∞ sin(nθ)
= sinθ dx = .
− x2 2xcosθ+1 n2
Z0 − n=1
X
The evaluations of Proposition 2 provide connections of infinite sums of differences
of Stieltjes constants with values Cl (θ).
2
6
The next result, among other relations, connects a sum of differences of Stieltjes
constants with a sum involving the sum of divisors function σ (n) = d 3. In
3 dn −
− |
stating the result, we introduce the Bloch-Wigner dilogarithm P
D(z) = Im[Li (z)+ln z ln(1 z)] = Im[Li (z)]+arg(1 z)ln z ,
2 2
| | − − | |
where, as usual for |z| ≤ 1, the dilogarithm function is given by Li2(z) = ∞n=1 nzn2,
which analytically continues to C (1, ). Then D(z) is real analytic in C P0,1 .
\ ∞ \{ }
Proposition 3 Let z = (1 + i√23)/2, z = 2 + i√23, z = (3 + i√23)/2, z =
1 2 3 4
(5+i√23)/2, and z = 3+i√23. Then
5
3 40π ζ(3) ∞
ζ(4)+ + σ (n)(1+πn√23)e πn√3
3 −
2 233/2 " 2 − #
n=1
X
4π2/3
= [21D(z )+7D(z )+D(z ) 3D(z )+D(z )]
233/2 1 2 3 − 4 5
∞ ( 1)n 22 j
= 23 2 − χ (j)γ
− 23 n
n! − 23
n=0 j=1 (cid:18) (cid:19)
X X
∞ ( 1)n 1 2 3 4 5 6
23 2 − γ +γ +γ +γ γ +γ
− n n n n n n
≡ n! 23 23 23 23 − 23 23
n=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
7 8 9 10 11 12 13 14
γ +γ +γ γ +γ +γ γ +γ
n n n n n n n n
− 23 23 23 − 23 23 23 − 23 23
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
15 16 17 18 19 20 21 22
γ +γ γ +γ γ +γ γ γ .
n n n n n n n n
− 23 23 − 23 23 − 23 23 − 23 − 23
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Proposition 4 (a)
1 ∞ ( 1)n k m
L (3) = − 2n χ (m)γ ,
±k k3 n! k n k
Xn=0 mX=1 (cid:16) (cid:17)
7
and (b) for all D, the values L (2) may similarly be written in terms of sums of
D
differences γ (a). In particular, for D < 0, closed-form expressions for L (3) are
k D
known.
After the proof of Proposition 4 we illustrate the use of an integral representation
for γ (a) in order to obtain such sums for rational values of a.
k
Proposition 5
∞ ( 1)n 1 3 1 1 π4
4 4 − 3n γ γ = ψ .
− n n ′′′
n! 4 − 4 768 4 − 96
n=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (cid:18) (cid:19)
X
The next sort of result follows from consideration of the Euler-Kronecker constant
γ of the quadratic field Q(√D). It provides a new summation representation of the
D
difference of the first Stieltjes constant at rational arguments.
Proposition 6 (a)
3 1 π2 ∞ 1 1
γ γ = +πγ +4π ,
1 4 − 1 4 3 ℓ(e2πℓ 1)
(cid:18) (cid:19) (cid:18) (cid:19) ℓ=1 −
X
and (b)
2 1 π π ∞ ( 1)ℓ 1
γ γ = +γ 4 − .
1 1
3 − 3 √3 "2√3 − ℓ [( 1)ℓ e√3πℓ]#
(cid:18) (cid:19) (cid:18) (cid:19) ℓ=1 − −
X
As we mention below, such differences of γ (a) at rational argument are related
1
to values of lnΓ(a) and hence lnΓ(1 a). For further details [13] may be consulted.
−
Proposition 7 (a)
p2 +1 3 ∞ ( 1)k 1 2
= − γ γ ,
p2 1 2π2 k! k 3 − k 3
p≡1 − k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Y X
mod 3
8
4 p2 1 1 ∞ ( 1)k 1 2
− = − γ γ ,
k k
27 p2 +1 9 k! 3 − 3
p≡2 k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
mYod 3 X
(b) (1.4b) holds, as well as
p2 +1 12 4 ∞ 1 1 3
= G = γ γ ,
k k
p2 1 π2 3π2 k! 4 − 4
p≡1 − k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
mYod 4 X
(c)
1 p2 1 p2 1
= − −
√5 p2 +1 p2 +1
p≡2 p≡3
mYod 5 mYod 5
∞ ( 1)k 1 2 3 4
= − γ γ γ +γ ,
k k k k
k! 5 − 5 − 5 5
k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
124 p3 1 p3 1
ζ(3) = − −
125 p3 +1 p3 +1
p≡2 p≡3
mYod 5 mYod 5
1 ∞ ( 2)k 1 2 3 4
= − γ γ γ +γ
k k k k
125 k! 5 − 5 − 5 5
k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
1 2 1 3 4
= ψ ψ +ψ ψ ,
′′ ′′ ′′ ′′
125 5 − 5 5 − 5
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(d)
3π3 7 p3 1 p3 1
= ζ(3) − −
64√2 8 p3 +1 p3 +1
p≡5 p≡7
mYod 8 mYod 8
1 ∞ ( 2)k 1 3 5 7
= − γ +γ γ γ ,
k k k k
83 k! 8 8 − 8 − 8
k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
π2 p2 1 p2 1
= 6ζ(2) − −
√2 p2 +1 p2 +1
p≡3 p≡5
Y Y
mod 8 mod 8
1 ∞ ( 1)k 1 3 5 7
= − γ γ γ +γ ,
k k k k
8 k! 8 − 8 − 8 8
k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
9
and (e)
∞ ( 1)k 1 3 5 7 45 ζ(4) p2 1 2
− γ +γ γ γ = − ,
k k k k
k! 8 8 − 8 − 8 32√2G p2 +1
k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) p≡7 (cid:18) (cid:19)
X Y
mod 8
∞ ( 2)k 1 3 5 7 3π6 1 p3 1 2
− γ γ γ +γ = − .
k k k k
k! 8 − 8 − 8 8 1792√2ζ(3) p3 +1
k=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) p≡7 (cid:18) (cid:19)
X Y
mod 8
Let E denotetheEuler numbers, forwhich theinitial valuesareE = 1, E = 1,
j 0 2
−
E = 5, E = 61, E = 1385, E = 50521, and E = 0 for n 0.
4 6 8 10 2n+1
− − ≥
The next result complements Proposition 2 in the case that D = 4.
−
Proposition 8 Let k 0. Then
≥
( 1)k π 2k+1 ∞ ( 1)n 1 3
− E = − γ γ (2k)n.
2k n n
2(2k)! 2 n! 4 − 4
(cid:16) (cid:17) Xn=0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Proposition 9 Sierpinski’s constant [28] S = γ 0.8228252496 has the repre-
4
− ≃
sentation S = 2γ + 4J , wherein
π 2
1 ln( lnx)
J − dx
2 ≡ 1+x2
Z0
has the novel representation for 0 < b < π/2
1 ∞ E2kb2k+1 b b 1 ∞ lnu
J = lnb + (lnb 1)+ du.
2
2 (2k +1)! − (2k +1) 2 − 2 coshu
k=1 (cid:20) (cid:21) Zb
X
In particular,
1 ∞ E2k 1 1 ∞ lnu
J = 1+ + du.
2
−2 (2k +1)!(2k +1) 2 coshu
" k=1 # Z1
X
Proposition 10. Let σ (n) be the sum of divisors function and K (z) the zeroth
1 0
order modified Bessel function of the second kind. Then (a)
1 1 √11 ∞
ζ L = γ +ln +4 ( 1)nσ (n)K (√11nπ)
11 1 0
2 − 2 8π ! −
(cid:18) (cid:19) (cid:18) (cid:19) n=1
X
10