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25 Zeta and Related FunctionsApplications

§25.16 Mathematical Applications

Contents
  1. §25.16(i) Distribution of Primes
  2. §25.16(ii) Euler Sums

§25.16(i) Distribution of Primes

In studying the distribution of primes px, Chebyshev (1851) introduced a function ψ(x) (not to be confused with the digamma function used elsewhere in this chapter), given by

25.16.1 ψ(x)=m=1pmxlnp,

which is related to the Riemann zeta function by

25.16.2 ψ(x)=xζ(0)ζ(0)ρxρρ+o(1),
x,

where the sum is taken over the nontrivial zeros ρ of ζ(s).

The prime number theorem (27.2.3) is equivalent to the statement

25.16.3 ψ(x)=x+o(x),
x.

The Riemann hypothesis is equivalent to the statement

25.16.4 ψ(x)=x+O(x12+ϵ),
x,

for every ϵ>0.

§25.16(ii) Euler Sums

Euler sums have the form

25.16.5 H(s)=n=1Hnns,

where Hn is given by (25.11.33).

H(s) is analytic for s>1, and can be extended meromorphically into the half-plane s>2k for every positive integer k by use of the relations

25.16.6 H(s)=ζ(s)+γζ(s)+12ζ(s+1)+r=1kζ(12r)ζ(s+2r)+n=11nsnB~2k+1(x)x2k+2dx,
25.16.7 H(s)=12ζ(s+1)+ζ(s)s1r=1k(s+2r22r1)ζ(12r)ζ(s+2r)(s+2k2k+1)n=11nnB~2k+1(x)xs+2k+1dx.

For integer s (2), H(s) can be evaluated in terms of the zeta function:

25.16.8 H(2) =2ζ(3),
H(3) =54ζ(4),
25.16.9 H(a)=a+22ζ(a+1)12r=1a2ζ(r+1)ζ(ar),
a=2,3,4,.

Also,

25.16.10 H(2a)=12ζ(12a)=B2a4a,
a=1,2,3,.

H(s) has a simple pole with residue ζ(12r) (=B2r/(2r)) at each odd negative integer s=12r, r=1,2,3,.

H(s) is the special case H(s,1) of the function

25.16.11 H(s,z)=n=11nsm=1n1mz,
(s+z)>1,

which satisfies the reciprocity law

25.16.12 H(s,z)+H(z,s)=ζ(s)ζ(z)+ζ(s+z),

when both H(s,z) and H(z,s) are finite.

For further properties of H(s,z) see Apostol and Vu (1984). Related results are:

25.16.13 n=1(Hnn)2 =174ζ(4),
25.16.14 r=1k=1r1rk(r+k) =54ζ(3),
25.16.15 r=1k=1r1r2(r+k) =34ζ(3).

For further generalizations, see Flajolet and Salvy (1998).