Particular values of the Riemann zeta function

The Riemann zeta function at 0 and 1

At zero, one has

ζ ( 0 ) = B 1 − = − B 1 + = − 1 2

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

lim ε → 0 ± ζ ( 1 + ε ) = ± ∞

Since it is a pole of first order, its principal value exists and is equal to the Euler–Mascheroni constant γ = 0.57721 56649+.

Even positive integers

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

ζ ( 2 n ) = ( − 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) !

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

ζ ( 2 ) = 1 + 1 2 2 + 1 3 2 + ⋯ = π 2 6 = 1.6449 … ( A013661) ζ ( 4 ) = 1 + 1 2 4 + 1 3 4 + ⋯ = π 4 90 = 1.0823 … ( A013662) ζ ( 6 ) = 1 + 1 2 6 + 1 3 6 + ⋯ = π 6 945 = 1.0173... … ( A013664) ζ ( 8 ) = 1 + 1 2 8 + 1 3 8 + ⋯ = π 8 9450 = 1.00407... … ( A013666) ζ ( 10 ) = 1 + 1 2 10 + 1 3 10 + ⋯ = π 10 93555 = 1.000994... … ( A013668) ζ ( 12 ) = 1 + 1 2 12 + 1 3 12 + ⋯ = 691 π 12 638512875 = 1.000246 … ( A013670) ζ ( 14 ) = 1 + 1 2 14 + 1 3 14 + ⋯ = 2 π 14 18243225 = 1.0000612 … ( A013672).

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

A n ζ ( n ) = B n π n

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

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

ζ ( 2 n ) = ∑ ℓ = 1 ∞ 1 ℓ 2 n = η n π 2 n

then we find recursively,

η 1 = 1 / 6 η n = ∑ ℓ = 1 n − 1 ( − 1 ) ℓ − 1 η n − ℓ ( 2 ℓ + 1 ) ! + ( − 1 ) n + 1 n ( 2 n + 1 ) !

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

Also, there is another recurrence:

ζ ( 2 n ) = 1 n + 1 2 ∑ k = 1 n − 1 ζ ( 2 k ) ζ ( 2 n − 2 k ) , n > 1

which can be proved, using that d d x cot ⁡ ( x ) = − 1 − cot 2 ⁡ ( x )

The values of the zeta function at non-negative even integers have the generating function:

∑ n = 0 ∞ ζ ( 2 n ) x 2 n = − π x 2 cot ⁡ ( π x ) = − 1 2 + π 2 6 x 2 + π 4 90 x 4 + π 6 945 x 6 + ⋯

Since

lim n → ∞ ζ ( 2 n ) = 1

The formula also shows that for n ∈ N , n → ∞ ,

| B 2 n | ∼ 2 ( 2 n ) ! ( 2 π ) 2 n

Odd positive integers

For the first few odd natural numbers one has

ζ ( 1 ) = 1 + 1 2 + 1 3 + ⋯ = ∞ ζ ( 3 ) = 1 + 1 2 3 + 1 3 3 + ⋯ = 1.20205 … ζ ( 5 ) = 1 + 1 2 5 + 1 3 5 + ⋯ = 1.03692 … ( A013663) ζ ( 7 ) = 1 + 1 2 7 + 1 3 7 + ⋯ = 1.00834 … ( A013665) ζ ( 9 ) = 1 + 1 2 9 + 1 3 9 + ⋯ = 1.002008 … ( 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.

The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic xxx spin chain.

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

ζ ( 5 ) = 1 294 π 5 − 72 35 ∑ n = 1 ∞ 1 n 5 ( e 2 π n − 1 ) − 2 35 ∑ n = 1 ∞ 1 n 5 ( e 2 π n + 1 ) ζ ( 5 ) = 12 ∑ n = 1 ∞ 1 n 5 sinh ⁡ ( π n ) − 39 20 ∑ n = 1 ∞ 1 n 5 ( e 2 π n − 1 ) − 1 20 ∑ n = 1 ∞ 1 n 5 ( e 2 π n + 1 )

ζ(7)

ζ ( 7 ) = 19 56700 π 7 − 2 ∑ n = 1 ∞ 1 n 7 ( e 2 π n − 1 )

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

ζ(2n + 1)

By defining the quantities

S ± ( s ) = ∑ n = 1 ∞ 1 n s ( e 2 π n ± 1 )

a series of relationships can be given in the form

0 = A n ζ ( n ) − B n π 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.

A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.

Negative integers

In general, for negative integers including zero, one has

ζ ( − n ) = ( − 1 ) n B n + 1 n + 1

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

ζ ( − 2 n ) = 0

The first few values for negative odd integers are

ζ ( − 1 ) = − 1 12 ζ ( − 3 ) = 1 120 ζ ( − 5 ) = − 1 252 ζ ( − 7 ) = 1 240 ζ ( − 9 ) = − 1 132 ζ ( − 11 ) = 691 32760 ζ ( − 13 ) = − 1 12

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

ζ ′ ( − 2 n ) = ( − 1 ) n ( 2 n ) ! 2 ( 2 π ) 2 n ζ ( 2 n + 1 )

The first few values of which are

ζ ′ ( − 2 ) = − ζ ( 3 ) 4 π 2 ζ ′ ( − 4 ) = 3 4 π 4 ζ ( 5 ) ζ ′ ( − 6 ) = − 45 8 π 6 ζ ( 7 ) ζ ′ ( − 8 ) = 315 4 π 8 ζ ( 9 )

One also has

ζ ′ ( 0 ) = − 1 2 ln ⁡ ( 2 π ) ≈ − 0.918938533 … ( A075700), ζ ′ ( − 1 ) = 1 12 − ln ⁡ A ≈ − 0.1654211437 … ( A084448)

and

ζ ′ ( 2 ) = 1 6 π 2 ( γ + ln ⁡ 2 − 12 ln ⁡ A + ln ⁡ π ) ≈ − 0.93754825 … ( A073002)

where A is the Glaisher–Kinkelin constant.

Series involving ζ(n)

The following sums can be derived from the generating function:

∑ k = 2 ∞ ζ ( k ) x k − 1 = − ψ 0 ( 1 − x ) − γ

where ψ₀ is the digamma function.

∑ k = 2 ∞ ( ζ ( k ) − 1 ) = 1 ∑ k = 1 ∞ ( ζ ( 2 k ) − 1 ) = 3 4 ∑ k = 1 ∞ ( ζ ( 2 k + 1 ) − 1 ) = 1 4 ∑ k = 2 ∞ ( − 1 ) k ( ζ ( k ) − 1 ) = 1 2

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

∑ k = 2 ∞ ( − 1 ) k ζ ( k ) k = γ ∑ k = 2 ∞ ζ ( k ) − 1 k = 1 − γ ∑ k = 2 ∞ ( − 1 ) k ζ ( k ) − 1 k = ln ⁡ 2 + γ − 1

and using the principal value

ζ ( k ) = lim ε → 0 ζ ( k + ε ) + ζ ( k − ε ) 2

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

∑ k = 1 ∞ ( − 1 ) k ζ ( k ) k = 0 ∑ k = 1 ∞ ζ ( k ) − 1 k = 0 ∑ k = 1 ∞ ( − 1 ) k ζ ( k ) − 1 k = ln ⁡ 2

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

Nontrivial zeros

Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". See Andrew Odlyzko's website for their tables and bibliographies.