25.15 Dirichlet \mathop{L\/}\nolimits-functions25.17 Physical Applications

§25.16 Mathematical Applications

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§25.16(i) Distribution of Primes

In studying the distribution of primes p\leq x, Chebyshev (1851) introduced a function \mathop{\psi\/}\nolimits\!\left(x\right) (not to be confused with the digamma function used elsewhere in this chapter), given by

25.16.1 \mathop{\psi\/}\nolimits\!\left(x\right)=\sum _{{m=1}}^{\infty}\sum _{{p^{m}\leq x}}\mathop{\ln\/}\nolimits p,

which is related to the Riemann zeta function by

25.16.2 \mathop{\psi\/}\nolimits\!\left(x\right)=x-\frac{{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(0\right)}{\mathop{\zeta\/}\nolimits\!\left(0\right)}-\sum _{\rho}\frac{x^{\rho}}{\rho}+\mathop{o\/}\nolimits\!\left(1\right), x\to\infty,

where the sum is taken over the nontrivial zeros \rho of \mathop{\zeta\/}\nolimits\!\left(s\right).

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

25.16.3 \mathop{\psi\/}\nolimits\!\left(x\right)=x+\mathop{o\/}\nolimits\!\left(x\right), x\to\infty.

The Riemann hypothesis is equivalent to the statement

25.16.4 \mathop{\psi\/}\nolimits\!\left(x\right)=x+\mathop{O\/}\nolimits\!\left(x^{{\frac{1}{2}+\epsilon}}\right), x\to\infty,

for every \epsilon>0.

§25.16(ii) Euler Sums

Euler sums have the form

25.16.5 \mathop{H\/}\nolimits\!\left(s\right)=\sum _{{n=1}}^{\infty}\frac{h(n)}{n^{s}},

where h(n) is given by (25.11.33).

\mathop{H\/}\nolimits\!\left(s\right) is analytic for \realpart{s}>1, and can be extended meromorphically into the half-plane \realpart{s}>-2k for every positive integer k by use of the relations

For integer s (\geq 2), \mathop{H\/}\nolimits\!\left(s\right) can be evaluated in terms of the zeta function:

25.16.8
\mathop{H\/}\nolimits\!\left(2\right)=2\mathop{\zeta\/}\nolimits\!\left(3\right),
\mathop{H\/}\nolimits\!\left(3\right)=\tfrac{5}{4}\mathop{\zeta\/}\nolimits\!\left(4\right),
25.16.9 \mathop{H\/}\nolimits\!\left(a\right)=\frac{a+2}{2}\mathop{\zeta\/}\nolimits\!\left(a+1\right)-\frac{1}{2}\sum _{{r=1}}^{{a-2}}\mathop{\zeta\/}\nolimits\!\left(r+1\right)\mathop{\zeta\/}\nolimits\!\left(a-r\right), a=2,3,4,\dots.

\mathop{H\/}\nolimits\!\left(s\right) has a simple pole with residue \mathop{\zeta\/}\nolimits\!\left(1-2r\right) (=-\mathop{B_{{2r}}\/}\nolimits/(2r)) at each odd negative integer s=1-2r, r=1,2,3,\dots.

\mathop{H\/}\nolimits\!\left(s\right) is the special case \mathop{H\/}\nolimits\!\left(s,1\right) of the function

25.16.11 \mathop{H\/}\nolimits\!\left(s,z\right)=\sum _{{n=1}}^{\infty}\frac{1}{n^{s}}\sum _{{m=1}}^{n}\frac{1}{m^{z}}, \realpart{(s+z)}>1,

which satisfies the reciprocity law

25.16.12 \mathop{H\/}\nolimits\!\left(s,z\right)+\mathop{H\/}\nolimits\!\left(z,s\right)=\mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\zeta\/}\nolimits\!\left(z\right)+\mathop{\zeta\/}\nolimits\!\left(s+z\right),

when both \mathop{H\/}\nolimits\!\left(s,z\right) and \mathop{H\/}\nolimits\!\left(z,s\right) are finite.

For further properties of \mathop{H\/}\nolimits\!\left(s,z\right) see Apostol and Vu (1984). Related results are:

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