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.

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

:$zeta(1)=infty.,$

Positive integers

Even positive integers

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

:$zeta(2n)\; =\; (-1)^\{n+1\}frac\{B\_\{2n\}(2pi)^\{2n\{2(2n)!\}$

for $nge\; 1$. The first few values are given by

:$zeta(2)\; =\; 1\; +\; frac\{1\}\{2^2\}\; +\; frac\{1\}\{3^2\}\; +\; cdots\; =\; frac\{pi^2\}\{6\}\; =\; 1.6449dots$; the demonstration of this equality is known as the Basel problem.:$zeta(4)\; =\; 1\; +\; frac\{1\}\{2^4\}\; +\; frac\{1\}\{3^4\}\; +\; cdots\; =\; frac\{pi^4\}\{90\}\; =\; 1.0823dots$; Stefan–Boltzmann law and Wien approximation in physics.:$zeta(6)\; =\; 1\; +\; frac\{1\}\{2^6\}\; +\; frac\{1\}\{3^6\}\; +\; cdots\; =\; frac\{pi^6\}\{945\}\; =\; 1.0173...dots$:$zeta(8)\; =\; 1\; +\; frac\{1\}\{2^8\}\; +\; frac\{1\}\{3^8\}\; +\; cdots\; =\; frac\{pi^8\}\{9450\}\; =\; 1.00407...\; dots$:$zeta(10)\; =\; 1\; +\; frac\{1\}\{2^\{10\; +\; frac\{1\}\{3^\{10\; +\; cdots\; =\; frac\{pi^\{10\{93555\}\; =\; 1.000994...dots$:$zeta(12)\; =\; 1\; +\; frac\{1\}\{2^\{12\; +\; frac\{1\}\{3^\{12\; +\; cdots\; =\; frac\{691pi^\{12\{638512875\}\; =\; 1.000246dots$:$zeta(14)\; =\; 1\; +\; frac\{1\}\{2^\{14\; +\; frac\{1\}\{3^\{14\; +\; cdots\; =\; frac\{2pi^\{14\{18243225\}\; =\; 1.0000612dots$

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 $A\_n$ and $B\_n$ are integers for all even "n". These are given by the integer sequences OEIS2C|id=A046988 and OEIS2C|id=A002432 in OEIS. Some of these values are reproduced below:

If we let $eta\_n$ be the coefficient $B/A$ as above,:$zeta(2n)\; =\; sum\_\{ell=1\}^\{infty\}frac\{1\}\{ell^\{2n=eta\_npi^\{2n\},$then we find recursively,:$eta\_1\; =\; frac\{1\}\{6\};$:$eta\_n=sum\_\{ell=1\}^\{n-1\}(-1)^\{ell-1\}frac\{eta\_\{n-ell\{(2ell+1)!\}+(-1)^\{n+1\}frac\{n\}\{(2n+1)!\}.$

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

The sequence for even numbers can also be derived from the Laurent expansion of the cotangent function around 0:

:$frac\{pi\}\{2\}cot(pi\; x)\; =\; frac\{1\}\{2\}x^\{-1\}-frac\{pi^2\}\{6\}x\; -frac\{pi^4\}\{90\}\; x^3\; -\; frac\{pi^6\}\{945\}x^5\; +\; ...$

Odd positive integers

For the first few odd natural numbers one has

:$zeta(1)\; =\; 1\; +\; frac\{1\}\{2\}\; +\; frac\{1\}\{3\}\; +\; cdots\; =\; infty$; this is the harmonic series.:$zeta(3)\; =\; 1\; +\; frac\{1\}\{2^3\}\; +\; frac\{1\}\{3^3\}\; +\; cdots\; =\; 1.20205dots$; this is called Apéry's constant:$zeta(5)\; =\; 1\; +\; frac\{1\}\{2^5\}\; +\; frac\{1\}\{3^5\}\; +\; cdots\; =\; 1.03692dots$:$zeta(7)\; =\; 1\; +\; frac\{1\}\{2^7\}\; +\; frac\{1\}\{3^7\}\; +\; cdots\; =\; 1.00834dots$:$zeta(9)\; =\; 1\; +\; frac\{1\}\{2^9\}\; +\; frac\{1\}\{3^9\}\; +\; cdots\; =\; 1.002008dots$

It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2"n"+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.

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

:$zeta(5)=frac\{1\}\{294\}pi^5\; -frac\{72\}\{35\}\; sum\_\{n=1\}^infty\; frac\{1\}\{n^5\; (e^\{2pi\; n\}\; -1)\}-frac\{2\}\{35\}\; sum\_\{n=1\}^infty\; frac\{1\}\{n^5\; (e^\{2pi\; n\}\; +1)\}$

and :$zeta(5)=12\; sum\_\{n=1\}^infty\; frac\{1\}\{n^5\; sinh\; (pi\; n)\}-frac\{39\}\{20\}\; sum\_\{n=1\}^infty\; frac\{1\}\{n^5\; (e^\{2pi\; n\}\; -1)\}-frac\{1\}\{20\}\; sum\_\{n=1\}^infty\; frac\{1\}\{n^5\; (e^\{2pi\; n\}\; +1)\}$

ζ(7)

:$zeta(7)=frac\{19\}\{56700\}pi^7\; -2\; sum\_\{n=1\}^infty\; frac\{1\}\{n^7\; (e^\{2pi\; n\}\; -1)\}$

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

ζ(2n+1)

By defining the quantities

:$S\_pm(s)\; =\; sum\_\{n=1\}^infty\; frac\{1\}\{n^s\; (e^\{2pi\; n\}\; pm\; 1)\}$

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:

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

Negative integers

In general, for negative integers, one has

:$zeta(-n)=-frac\{B\_\{n+1\{n+1\}$

for $nge\; 1$.

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 + · · ·.

Derivatives

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

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

The first few values of which are

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

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

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

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

One also has

:$zeta^\{prime\}(0)\; =\; -frac\{1\}\{2\}log(2pi)approx\; -0.918938533ldots$

and

:$zeta^\{prime\}(-1)=frac\{1\}\{12\}-log\; A\; approx\; -0.165421137ldots$

where $A$ is the Glaisher-Kinkelin constant.

um of Zeta Constants

:$sum\_\{k=2\}^infty\; (zeta(k)\; -1)\; =\; 1$

References

* Simon Plouffe, " [http://www.lacim.uqam.ca/~plouffe/identities.html Identities inspired from Ramanujan Notebooks] ", (1998).
* Simon Plouffe, " [http://www.lacim.uqam.ca/~plouffe/inspired22.html Identities inspired by Ramanujan Notebooks part 2] ] [http://www.lacim.uqam.ca/~plouffe/inspired2.pdf PDF] " (2006).
* Linas Vepstas, " [http://www.linas.org/math/plouffe-ram.pdf On Plouffe's Ramanujan Identities] ", ArXiv [http://arxiv.org/abs/Math.NT/0609775 Math.NT/0609775] (2006).
* Wadim Zudilin, "One of the Numbers ζ(5), ζ(7), ζ(9), ζ(11) Is Irrational." Uspekhi Mat. Nauk 56, 149-150, 2001. [http://wain.mi.ras.ru/PS/zeta5-11$.pdf PDF] [http://wain.mi.ras.ru/PS/zeta5-11$.ps.gz PS] [http://wain.mi.ras.ru/PS/zeta5-11.pdf PDF Russian] [http://wain.mi.ras.ru/PS/zeta5-11.ps.gz PS Russian]