information:classification
Dirichlet series generating functions (DGFs)

Formal Dirichlet series are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence a is: DG (a n; s) = ∑ n = 1 ∞ a n n s. {\displaystyle \operatorname {DG} (a_{n};s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}

The Dirichlet series generating function is especially useful when a is a multiplicative function, in which case it has an Euler product expression in terms of the function's Bell series: DG (a n; s) = ∏ p BG p (a n; p − s). {\displaystyle \operatorname {DG} (a_{n};s)=\prod _{p}\operatorname {BG} _{p}(a_{n};p^{-s})\,.}

If a is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L -series. We also have a relation between the pair of coefficients in the Lambert series expansions above and their DGFs. Namely, we can prove that: LG (a n; x) = b n {\displaystyle \operatorname {LG} (a_{n};x)=b_{n}} if and only if DG (a n; s) ζ (s) = DG (b n; s), {\displaystyle \operatorname {DG} (a_{n};s)\zeta (s)=\operatorname {DG} (b_{n};s),} where ζ (s) is the Riemann zeta function.

The sequence a generated by a Dirichlet series generating function (DGF) corresponding to: DG (a k; s) = ζ (s) m {\displaystyle \operatorname {DG} (a_{k};s)=\zeta (s)^{m}} has the ordinary generating function: ∑ k = 1 k = n a k x k = x + (m 1) ∑ 2 ≤ a ≤ n x a + (m 2) ∑ a = 2 ∞ ∑ b = 2 ∞ a b ≤ n x a b + (m 3) ∑ a = 2 ∞ ∑ c = 2 ∞ ∑ b = 2 ∞ a b c ≤ n x a b c + (m 4) ∑ a = 2 ∞ ∑ b = 2 ∞ ∑ c = 2 ∞ ∑ d = 2 ∞ a b c d ≤ n x a b c d + ⋯ {\displaystyle \sum _{k=1}^{k=n}a_{k}x^{k}=x+{\binom {m}{1}}\sum _{2\leq a\leq n}x^{a}+{\binom {m}{2}}{\underset {ab\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }}}x^{ab}+{\binom {m}{3}}{\underset {abc\leq n}{\sum _{a=2}^{\infty }\sum _{c=2}^{\infty }\sum _{b=2}^{\infty }}}x^{abc}+{\binom {m}{4}}{\underset {abcd\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }\sum _{c=2}^{\infty }\sum _{d=2}^{\infty }}}x^{abcd}+\cdots }