Zeta constant

In mathematics, a zeta constant is a value of the Riemann zeta function, with the argument being integral. This article provides a number of series identities for the zeta function for integer values.

The Riemann zeta function at 0 and 1

At zero, one has

$\zeta(0)= -B_1=-\tfrac{1}{2}.\!$

At 1 there is a pole, so ζ(1) is not defined but the left and right limits are:

$\lim_{\epsilon\to 0^{\pm}}\zeta(1+\epsilon) = \pm\infty$

and because it is a pole of 1st order its principle value exists and is γ.

Positive integers

Even positive integers

For the even positive integers, one has the relationship to the Bernoulli numbers:

$\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!} \!$

for n ∈ N. The first few values are given by:

$\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} = 1.6449\dots\!$ ( A013661)

(the demonstration of this equality is known as the Basel problem)

$\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} = 1.0823\dots\!$ ( A013662)

(the Stefan–Boltzmann law and Wien approximation in physics)

$\zeta(6) = 1 + \frac{1}{2^6} + \frac{1}{3^6} + \cdots = \frac{\pi^6}{945} = 1.0173...\dots\!$ ( A013664)

$\zeta(8) = 1 + \frac{1}{2^8} + \frac{1}{3^8} + \cdots = \frac{\pi^8}{9450} = 1.00407... \dots\!$ ( A013666)

$\zeta(10) = 1 + \frac{1}{2^{10}} + \frac{1}{3^{10}} + \cdots = \frac{\pi^{10}}{93555} = 1.000994...\dots\!$ ( A013668)

$\zeta(12) = 1 + \frac{1}{2^{12}} + \frac{1}{3^{12}} + \cdots = \frac{691\pi^{12}}{638512875} = 1.000246\dots\!$ ( A013670)

$\zeta(14) = 1 + \frac{1}{2^{14}} + \frac{1}{3^{14}} + \cdots = \frac{2\pi^{14}}{18243225} = 1.0000612\dots\!$ ( A013672).

The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as

$A_n \zeta(n) = B_n \pi^{n}\,\!$

where A_n and B_n are integers for all even n. These are given by the integer sequences  A046988 and  A002432 in OEIS. Some of these values are reproduced below:

coefficients

2n	A	B
2	6	1
4	90	1
6	945	1
8	9450	1
10	93555	1
12	638512875	691
14	18243225	2
16	325641566250	3617
18	38979295480125	43867
20	1531329465290625	174611
22	13447856940643125	155366
24	201919571963756521875	236364091
26	11094481976030578125	1315862
28	564653660170076273671875	6785560294
30	5660878804669082674070015625	6892673020804
32	62490220571022341207266406250	7709321041217
34	12130454581433748587292890625	151628697551

If we let η_n be the coefficient B/A as above,

$\zeta(2n) = \sum_{\ell=1}^{\infty}\frac{1}{\ell^{2n}}=\eta_n\pi^{2n},$

then we find recursively,

$\begin{align} \eta_1 &= 1/6; \\ \eta_n &= \sum_{\ell=1}^{n-1}(-1)^{\ell-1}\frac{\eta_{n-\ell}}{(2\ell+1)!}+(-1)^{n+1}\frac{n}{(2n+1)!}. \end{align}$

This recurrence relation may be derived from that for the Bernoulli numbers.

The even zeta constants have the generating function:

$\sum_{n=0}^\infty \zeta(2n) x^{2n} = -\frac{\pi x}{2} \cot(\pi x) = -\frac{1}{2} + \frac{\pi^2}{6} x^2 + \frac{\pi^4}{90} x^4+\frac{\pi^6}{945}x^6 + \cdots$

Since

$\lim_{n\rightarrow\infty} \zeta(2n)=1,$

the formula also shows that for $n\in\mathbb{N}, n\rightarrow\infty$,

$\left|B_{2n}\right| \sim \frac{2(2n)!}{(2\pi)^{2n}}$.

Odd positive integers

For the first few odd natural numbers one has

$\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty\!$

(the harmonic series);

$\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots = 1.20205\dots\!$

(Apéry's constant)

$\zeta(5) = 1 + \frac{1}{2^5} + \frac{1}{3^5} + \cdots = 1.03692\dots\!$  A013663

$\zeta(7) = 1 + \frac{1}{2^7} + \frac{1}{3^7} + \cdots = 1.00834\dots\!$  A013665

$\zeta(9) = 1 + \frac{1}{2^9} + \frac{1}{3^9} + \cdots = 1.002008\dots\!$  A013667

It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n+1) (n ∈ N) are irrational. There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ(5), ζ(7), ζ(9), or ζ(11) is irrational. This has significance for application to physics. Correlation functions in antiferromagnetic xxx spin chain can be expressed in terms of values of zeta at odd arguments ζ(2n+1).

Most of the identities following below are provided by Simon Plouffe. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.

ζ(5)

Plouffe gives the following identities

$\begin{align} \zeta(5)&=\frac{1}{294}\pi^5 -\frac{72}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{2}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}\\ \zeta(5)&=12 \sum_{n=1}^\infty \frac{1}{n^5 \sinh (\pi n)} -\frac{39}{20} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{1}{20} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)} \end{align}$

ζ(7)

$\zeta(7)=\frac{19}{56700}\pi^7 -2 \sum_{n=1}^\infty \frac{1}{n^7 (e^{2\pi n} -1)}\!$

Note that the sum is in the form of the Lambert series.

ζ(2n+1)

By defining the quantities

$S_\pm(s) = \sum_{n=1}^\infty \frac{1}{n^s (e^{2\pi n} \pm 1)}$

a series of relationships can be given in the form

$0=A_n \zeta(n) - B_n \pi^{n} + C_n S_-(n) + D_n S_+(n)\,$

where A_n, B_n, C_n and D_n are positive integers. Plouffe gives a table of values:

coefficients

n	A	B	C	D
3	180	7	360	0
5	1470	5	3024	84
7	56700	19	113400	0
9	18523890	625	37122624	74844
11	425675250	1453	851350500	0
13	257432175	89	514926720	62370
15	390769879500	13687	781539759000	0
17	1904417007743250	6758333	3808863131673600	29116187100
19	21438612514068750	7708537	42877225028137500	0
21	1881063815762259253125	68529640373	3762129424572110592000	1793047592085750

These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.

The only fast algorithm for the calculation of Riemann's zeta function for any integer argument was found by E.A. Karatsuba.

Negative integers

In general, for negative integers, one has

$\zeta(-n)=-\frac{B_{n+1}}{n+1}.$

The so-called "trivial zeros" occur at the negative even integers:

$\zeta(-2n)=0.\,$

The first few values for negative odd integers are

$\zeta(-1)=-\frac{1}{12}$

$\zeta(-3)=\frac{1}{120}$

$\zeta(-5)=-\frac{1}{252}$

$\zeta(-7)=\frac{1}{240}.$

However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on the first value, see 1 + 2 + 3 + 4 + · · ·.

So ζ(m) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.

Derivatives

The derivative of the zeta function at the negative even integers is given by

$\zeta^{\prime}(-2n) = (-1)^n \frac {(2n)!} {2 (2\pi)^{2n}} \zeta (2n+1).$

The first few values of which are

$\zeta^{\prime}(-2) = -\frac{\zeta(3)}{4\pi^2}$

$\zeta^{\prime}(-4) = \frac{3}{4\pi^4} \zeta(5)$

$\zeta^{\prime}(-6) = -\frac{45}{8\pi^6} \zeta(7)$

$\zeta^{\prime}(-8) = \frac{315}{4\pi^8} \zeta(9).$

One also has

$\zeta^{\prime}(0) = -\frac{1}{2}\ln(2\pi)\approx -0.918938533\ldots$  A075700

and

$\zeta^{\prime}(-1)=\frac{1}{12}-\ln A \approx -0.1654211437\ldots$  A084448

where A is the Glaisher-Kinkelin constant.

Sum of zeta constants

The following sums can be derived from the generating function:

$\sum_{k=2}^\infty \zeta(k) x^{k-1}=-\psi_0(1-x)-\gamma$

where ψ₀ is the digamma function.

$\sum_{k=2}^\infty (\zeta(k) -1) = 1$

$\sum_{k=1}^\infty (\zeta(2k) -1) = \frac{3}{4}$

$\sum_{k=1}^\infty (\zeta(2k+1) -1) = \frac{1}{4}$

$\sum_{k=2}^\infty (-1)^k(\zeta(k) -1) = \frac{1}{2}.$

Series related to the Euler–Mascheroni constant (denoted by γ) are

$\sum_{k=2}^\infty (-1)^k \frac{\zeta(k)}{k} = \gamma$

$\sum_{k=2}^\infty \frac{\zeta(k) - 1}{k} = 1 - \gamma$

$\sum_{k=2}^\infty (-1)^k \frac{\zeta(k)-1}{k} = \ln2 + \gamma - 1$

and using the principle value

$\zeta(k) = \lim_{\varepsilon \to 0} \frac{\zeta(k+\varepsilon)+\zeta(k-\varepsilon)}{2},$

which of course affects only the value at 1. These formulae can be stated as

$\sum_{k=1}^\infty (-1)^k \frac{\zeta(k)}{k} = 0$

$\sum_{k=1}^\infty \frac{\zeta(k) - 1}{k} = 0$

$\sum_{k=1}^\infty (-1)^k \frac{\zeta(k)-1}{k} = \ln2$

and show that they depend on the principle value of ζ(1) = γ.

References==

• Karatsuba, E. A. (1995). "Fast calculation of the Rieman Zeta function zeta(s) for integer values of the argument s". Probl. Perdachi Inf. 31 (4): 69–80. MR 1367927. 
• Simon Plouffe, "Identities inspired from Ramanujan Notebooks", (1998).
• Simon Plouffe, "Identities inspired by Ramanujan Notebooks part 2 PDF" (2006).
• Vepstas, Linas (2006). "On Plouffe's Ramanujan Identities". arXiv:math.NT/0609775.
• Zudilin, Wadim (2001). "One of the Numbers ζ(5), ζ(7), ζ(9), ζ(11) Is Irrational". Russian Mathematical Surveys 56: 774–776. doi:10.1070/RM2001v056n04ABEH000427. MR 1861452.  PDF PDF Russian PS Russian