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25 Zeta and Related FunctionsApplications

§25.16 Mathematical Applications


§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)=x-ζ(0)ζ(0)-ρxρρ+o(1),

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),

The Riemann hypothesis is equivalent to the statement

25.16.4 ψ(x)=x+O(x12+ϵ),

for every ϵ>0.

§25.16(ii) Euler Sums

Euler sums have the form

25.16.5 H(s)=n=1h(n)ns,

where h(n) 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(1-2r)ζ(s+2r)+n=11nsnB~2k+1(x)x2k+2dx,
25.16.7 H(s)=12ζ(s+1)+ζ(s)s-1-r=1k(s+2r-22r-1)ζ(1-2r)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=1a-2ζ(r+1)ζ(a-r),


25.16.10 H(-2a)=12ζ(1-2a)=-B2a4a,

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

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(h(n)n)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).