Xrthnty

This article gives some specific values of the Riemann zeta function, including values at integer arguments, and some series involving them.

The Riemann zeta function at 0 and 1

At zero, one has

ζ(0)=B1−=−B1+=−12displaystyle zeta (0)=B_1^-=-B_1^+=-tfrac 12!

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

limε→0±ζ(1+ε)=±∞displaystyle lim _varepsilon to 0^pm zeta (1+varepsilon )=pm infty

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

Positive integers

Even positive integers

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

ζ(2n)=(−1)n+1B2n(2π)2n2(2n)!displaystyle zeta (2n)=(-1)^n+1frac B_2n(2pi )^2n2(2n)!!

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

ζ(2)=1+122+132+⋯=π26=1.6449…displaystyle zeta (2)=1+frac 12^2+frac 13^2+cdots =frac pi ^26=1.6449dots ! (OEIS: A013661)

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

ζ(4)=1+124+134+⋯=π490=1.0823…displaystyle zeta (4)=1+frac 12^4+frac 13^4+cdots =frac pi ^490=1.0823dots ! (OEIS: A013662)

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

ζ(6)=1+126+136+⋯=π6945=1.0173...…displaystyle zeta (6)=1+frac 12^6+frac 13^6+cdots =frac pi ^6945=1.0173...dots ! (OEIS: A013664)

ζ(8)=1+128+138+⋯=π89450=1.00407...…displaystyle zeta (8)=1+frac 12^8+frac 13^8+cdots =frac pi ^89450=1.00407...dots ! (OEIS: A013666)

ζ(10)=1+1210+1310+⋯=π1093555=1.000994...…displaystyle zeta (10)=1+frac 12^10+frac 13^10+cdots =frac pi ^1093555=1.000994...dots ! (OEIS: A013668)

ζ(12)=1+1212+1312+⋯=691π12638512875=1.000246…displaystyle zeta (12)=1+frac 12^12+frac 13^12+cdots =frac 691pi ^12638512875=1.000246dots ! (OEIS: A013670)

ζ(14)=1+1214+1314+⋯=2π1418243225=1.0000612…displaystyle zeta (14)=1+frac 12^14+frac 13^14+cdots =frac 2pi ^1418243225=1.0000612dots ! (OEIS: A013672).

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

Anζ(2n)=Bnπ2ndisplaystyle A_nzeta (2n)=B_npi ^2n,!

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

If we let ηn=Bn/Andisplaystyle eta _n=B_n/A_n be the coefficient of π2ndisplaystyle pi ^2n as above,

ζ(2n)=∑ℓ=1∞1ℓ2n=ηnπ2ndisplaystyle zeta (2n)=sum _ell =1^infty frac 1ell ^2n=eta _npi ^2n

then we find recursively,

η1=1/6ηn=∑ℓ=1n−1(−1)ℓ−1ηn−ℓ(2ℓ+1)!+(−1)n+1n(2n+1)!displaystyle beginalignedeta _1&=1/6\eta _n&=sum _ell =1^n-1(-1)^ell -1frac eta _n-ell (2ell +1)!+(-1)^n+1frac n(2n+1)!endaligned

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

Also, there is another recurrence:

ζ(2n)=1n+12∑k=1n−1ζ(2k)ζ(2n−2k),n>1displaystyle zeta (2n)=frac 1n+frac 12sum _k=1^n-1zeta (2k)zeta (2n-2k),n>1

which can be proved, using that ddxcot⁡(x)=−1−cot2⁡(x)displaystyle frac ddxcot(x)=-1-cot ^2(x)

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

∑n=0∞ζ(2n)x2n=−πx2cot⁡(πx)=−12+π26x2+π490x4+π6945x6+⋯displaystyle sum _n=0^infty zeta (2n)x^2n=-frac pi x2cot(pi x)=-frac 12+frac pi ^26x^2+frac pi ^490x^4+frac pi ^6945x^6+cdots

Since

limn→∞ζ(2n)=1displaystyle lim _nrightarrow infty zeta (2n)=1

The formula also shows that for n∈N,n→∞displaystyle nin mathbb N ,nrightarrow infty ,

|B2n|∼2(2n)!(2π)2nsim frac 2(2n)!(2pi )^2n

Odd positive integers

For the first few odd natural numbers one has

ζ(1)=1+12+13+⋯=∞displaystyle zeta (1)=1+frac 12+frac 13+cdots =infty !

(the harmonic series);

ζ(3)=1+123+133+⋯=1.20205…displaystyle zeta (3)=1+frac 12^3+frac 13^3+cdots =1.20205dots ! (OEIS: A02117)

(Apéry's constant)

ζ(5)=1+125+135+⋯=1.03692…displaystyle zeta (5)=1+frac 12^5+frac 13^5+cdots =1.03692dots ! (OEIS: A013663)

ζ(7)=1+127+137+⋯=1.00834…displaystyle zeta (7)=1+frac 12^7+frac 13^7+cdots =1.00834dots ! (OEIS: A013665)

ζ(9)=1+129+139+⋯=1.002008…displaystyle zeta (9)=1+frac 12^9+frac 13^9+cdots =1.002008dots ! (OEIS: 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.^[1] 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.^[2]

The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.^[3]

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)=1294π5−7235∑n=1∞1n5(e2πn−1)−235∑n=1∞1n5(e2πn+1)ζ(5)=12∑n=1∞1n5sinh⁡(πn)−3920∑n=1∞1n5(e2πn−1)−120∑n=1∞1n5(e2πn+1)displaystyle beginalignedzeta (5)&=frac 1294pi ^5-frac 7235sum _n=1^infty frac 1n^5(e^2pi n-1)-frac 235sum _n=1^infty frac 1n^5(e^2pi n+1)\zeta (5)&=12sum _n=1^infty frac 1n^5sinh(pi n)-frac 3920sum _n=1^infty frac 1n^5(e^2pi n-1)-frac 120sum _n=1^infty frac 1n^5(e^2pi n+1)endaligned

ζ(7)

ζ(7)=1956700π7−2∑n=1∞1n7(e2πn−1)displaystyle zeta (7)=frac 1956700pi ^7-2sum _n=1^infty frac 1n^7(e^2pi 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∞1ns(e2πn±1)displaystyle S_pm (s)=sum _n=1^infty frac 1n^s(e^2pi npm 1)

a series of relationships can be given in the form

0=Anζ(n)−Bnπn+CnS−(n)+DnS+(n)displaystyle 0=A_nzeta (n)-B_npi ^n+C_nS_-(n)+D_nS_+(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.^[4][5][6]

Negative integers

In general, for negative integers (and also zero), one has

ζ(−n)=(−1)nBn+1n+1displaystyle zeta (-n)=(-1)^nfrac B_n+1n+1

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

ζ(−2n)=0displaystyle zeta (-2n)=0,

The first few values for negative odd integers are

ζ(−1)=−112ζ(−3)=1120ζ(−5)=−1252ζ(−7)=1240ζ(−9)=−1132ζ(−11)=69132760ζ(−13)=−112displaystyle beginalignedzeta (-1)&=-frac 112\zeta (-3)&=frac 1120\zeta (-5)&=-frac 1252\zeta (-7)&=frac 1240\zeta (-9)&=-frac 1132\zeta (-11)&=frac 69132760\zeta (-13)&=-frac 112endaligned

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′(−2n)=(−1)n(2n)!2(2π)2nζ(2n+1)displaystyle zeta ^prime (-2n)=(-1)^nfrac (2n)!2(2pi )^2nzeta (2n+1)

The first few values of which are

ζ′(−2)=−ζ(3)4π2ζ′(−4)=34π4ζ(5)ζ′(−6)=−458π6ζ(7)ζ′(−8)=3154π8ζ(9)displaystyle beginalignedzeta ^prime (-2)&=-frac zeta (3)4pi ^2\[6pt]zeta ^prime (-4)&=frac 34pi ^4zeta (5)\[6pt]zeta ^prime (-6)&=-frac 458pi ^6zeta (7)\[6pt]zeta ^prime (-8)&=frac 3154pi ^8zeta (9)endaligned

One also has

ζ′(0)=−12ln⁡(2π)≈−0.918938533…displaystyle zeta ^prime (0)=-frac 12ln(2pi )approx -0.918938533ldots (OEIS: A075700),

ζ′(−1)=112−ln⁡A≈−0.1654211437…displaystyle zeta ^prime (-1)=frac 112-ln Aapprox -0.1654211437ldots (OEIS: A084448)

and

ζ′(2)=16π2(γ+ln⁡2−12ln⁡A+ln⁡π)≈−0.93754825…displaystyle zeta ^prime (2)=frac 16pi ^2(gamma +ln 2-12ln A+ln pi )approx -0.93754825ldots (OEIS: 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)xk−1=−ψ0(1−x)−γdisplaystyle sum _k=2^infty zeta (k)x^k-1=-psi _0(1-x)-gamma

where ψ₀ is the digamma function.

∑k=2∞(ζ(k)−1)=1displaystyle sum _k=2^infty (zeta (k)-1)=1

∑k=1∞(ζ(2k)−1)=34displaystyle sum _k=1^infty (zeta (2k)-1)=frac 34

∑k=1∞(ζ(2k+1)−1)=14displaystyle sum _k=1^infty (zeta (2k+1)-1)=frac 14

∑k=2∞(−1)k(ζ(k)−1)=12displaystyle sum _k=2^infty (-1)^k(zeta (k)-1)=frac 12

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

∑k=2∞(−1)kζ(k)k=γdisplaystyle sum _k=2^infty (-1)^kfrac zeta (k)k=gamma

∑k=2∞ζ(k)−1k=1−γdisplaystyle sum _k=2^infty frac zeta (k)-1k=1-gamma

∑k=2∞(−1)kζ(k)−1k=ln⁡2+γ−1displaystyle sum _k=2^infty (-1)^kfrac zeta (k)-1k=ln 2+gamma -1

and using the principal value

ζ(k)=limε→0ζ(k+ε)+ζ(k−ε)2displaystyle zeta (k)=lim _varepsilon to 0frac zeta (k+varepsilon )+zeta (k-varepsilon )2

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

∑k=1∞(−1)kζ(k)k=0displaystyle sum _k=1^infty (-1)^kfrac zeta (k)k=0

∑k=1∞ζ(k)−1k=0displaystyle sum _k=1^infty frac zeta (k)-1k=0

∑k=1∞(−1)kζ(k)−1k=ln⁡2displaystyle sum _k=1^infty (-1)^kfrac zeta (k)-1k=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.

References

Further reading

• 
  Ciaurri, Óscar; Navas, Luis M.; Ruiz, Francisco J.; Varona, Juan L. (May 2015). "A Simple Computation of ζ(2k)". The American Mathematical Monthly. 122 (5): 444–451. doi:10.4169/amer.math.monthly.122.5.444. JSTOR 10.4169/amer.math.monthly.122.5.444.
  


• 
  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" (PDF). 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. Bibcode:2001RuMaS..56..774Z. doi:10.1070/RM2001v056n04ABEH000427. MR 1861452.
  PDF PDF Russian PS Russian
  


• Nontrival zeros reference by Andrew Odlyzko:
  
  • Bibliography
  
  
  • Tables