Zeta constant

In mathematics, a zeta constant is a number obtained by plugging an integer into the Riemann zeta function. This article provides a number of series identities for the zeta function for integer values.

Contents

The Riemann zeta function at 0 and 1

At zero, one has

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

There is a pole at 1, so \zeta(1) is not defined but the left and right limits are:

\lim_{\epsilon\to 0^%2B}\zeta(1%2B\epsilon) = \infty

and \lim_{\epsilon\to 0^-}\zeta(1%2B\epsilon) = -\infty

Positive integers

Even positive integers

For the even positive integers, one has the well-known relationship to the Bernoulli numbers, given by Euler:

\zeta(2n) = (-1)^{n%2B1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!} \!

for n \in \mathbb{N}. The first few values are given by:

\zeta(2) = 1 %2B \frac{1}{2^2} %2B \frac{1}{3^2} %2B \cdots = \frac{\pi^2}{6} = 1.6449\dots\! ( A013661)
(the demonstration of this equality is known as the Basel problem)
\zeta(4) = 1 %2B \frac{1}{2^4} %2B \frac{1}{3^4} %2B \cdots = \frac{\pi^4}{90} = 1.0823\dots\! ( A013662)
(the Stefan–Boltzmann law and Wien approximation in physics)
\zeta(6) = 1 %2B \frac{1}{2^6} %2B \frac{1}{3^6} %2B \cdots = \frac{\pi^6}{945} = 1.0173...\dots\! ( A013664)
\zeta(8) = 1 %2B \frac{1}{2^8} %2B \frac{1}{3^8} %2B \cdots = \frac{\pi^8}{9450} = 1.00407... \dots\! ( A013666)
\zeta(10) = 1 %2B \frac{1}{2^{10}} %2B \frac{1}{3^{10}} %2B \cdots = \frac{\pi^{10}}{93555} = 1.000994...\dots\! ( A013668)
\zeta(12) = 1 %2B \frac{1}{2^{12}} %2B \frac{1}{3^{12}} %2B \cdots = \frac{691\pi^{12}}{638512875} = 1.000246\dots\! ( A013670)
\zeta(14) = 1 %2B \frac{1}{2^{14}} %2B \frac{1}{3^{14}} %2B \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

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

where An and Bn 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 \eta_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,

\eta_1 = 1/6;\!
\eta_n=\sum_{\ell=1}^{n-1}(-1)^{\ell-1}\frac{\eta_{n-\ell}}{(2\ell%2B1)!}%2B(-1)^{n%2B1}\frac{n}{(2n%2B1)!}.

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} %2B \frac{\pi^2}{6} x^2 %2B \frac{\pi^4}{90} x^4%2B\frac{\pi^6}{945}x^6 %2B \cdots

Since \lim_{n\rightarrow\infty} \zeta(2n)=1, the formula also shows that for n\in\mathbb{N}:

\left|B_{2n}\right|\sim\frac{2(2n)!}{(2\pi)^{2n}} as n\rightarrow\infty

where the symbol ~ means "asymptotic to" (see asymptotic notation).

Odd positive integers

For the first few odd natural numbers one has

\zeta(1) = 1 %2B \frac{1}{2} %2B \frac{1}{3} %2B \cdots = \infty\! (the harmonic series);
\zeta(3) = 1 %2B \frac{1}{2^3} %2B \frac{1}{3^3} %2B \cdots = 1.20205\dots\! (Apéry's constant)
\zeta(5) = 1 %2B \frac{1}{2^5} %2B \frac{1}{3^5} %2B \cdots = 1.03692\dots\!  A013663
\zeta(7) = 1 %2B \frac{1}{2^7} %2B \frac{1}{3^7} %2B \cdots = 1.00834\dots\!  A013665
\zeta(9) = 1 %2B \frac{1}{2^9} %2B \frac{1}{3^9} %2B \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 (ir)rationality 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 argumets \zeta(2n%2B1).

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 identities

\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} %2B1)}\!

and

\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} %2B1)}\!

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} %2B C_n S_-(n) %2B D_n S_%2B(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 calculation of Riemann's zeta function for any integer argument was found by E.A. Karatsuba[1][2][3]

Negative integers

In general, for negative integers, one has

\zeta(-n)=-\frac{B_{n%2B1}}{n%2B1}

for n\in\mathbb{N}.

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 + · · ·.

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%2B1), n\in\mathbb{N}.

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 \psi_0 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%2B1) -1) = \frac{1}{4}
\sum_{k=2}^\infty (-1)^n(\zeta(k) -1) = \frac{1}{2}

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

\sum_{k=2}^\infty (-1)^n \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 %2B \gamma - 1

References

  1. ^ E.A. Karatsuba: Fast computation of the Riemann zeta-function ζ(s) for integer values of the argument s. Probl. Inf. Transm. Vol.31, No.4, pp.353-362 (1995).
  2. ^ E.A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p.626 (1996).
  3. ^ E.A. Karatsuba: Fast evaluation of ζ(3). Probl. Inf. Transm. Vol.29, No.1, pp.58-62 (1993).