Zeta functions of graphs : a stroll through the garden , by Audrey Terras , Cambridge

1.2. Dirichlet. A sequence of numbers a1, a2, . . . gives rise to a Dirichlet series f(s) = ∑ n≥1 ann −s. This is but a method of encoding the infinitely many numbers (an) into a single mathematical object f . If the numbers an grow at most polynomially, then f will actually be an analytic function on a half-plane { (s) > λ}; and the abscissa of convergence λ is related to the growth of the sequence in that a1 + · · · + an n. Much finer asymptotic information on the sequence (e.g., in the form of an approximation an ∝ n(log n)) can be obtained by finer analytic properties of f . This goes under the headline of Abelian and Tauberian theorems. Gustav Lejeune Dirichlet (1805–1859) did much more than get his name attributed to a general form of formal series; he considered expressions of the form ∑ n≥1 χ(n)n −s for a multiplicative character χ of Z/NZ. Such expressions are