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25 Zeta and Related FunctionsNotation

§25.1 Special Notation

(For other notation see Notation for the Special Functions.)

k,m,n nonnegative integers.
p prime number.
x real variable.
a real or complex parameter.
s=σ+it complex variable.
z=x+iy complex variable.
γ Euler’s constant (§5.2(ii)).
ψ(x) digamma function Γ(x)/Γ(x) except in §25.16. See §5.2(i).
Bn,Bn(x) Bernoulli number and polynomial (§24.2(i)).
B~n(x) periodic Bernoulli function Bn(x-x).
m|n m divides n.
primes on function symbols: derivatives with respect to argument.

The main function treated in this chapter is the Riemann zeta function ζ(s). This notation was introduced in Riemann (1859).

The main related functions are the Hurwitz zeta function ζ(s,a), the dilogarithm Li2(z), the polylogarithm Lis(z) (also known as Jonquière’s function ϕ(z,s)), Lerch’s transcendent Φ(z,s,a), and the Dirichlet L-functions L(s,χ).