a Dirichlet series
/ deer-ish-LAY or DEER-ish-lay /
A Dirichlet series is the natural generalisation of the zeta function's defining sum. It has the shape sum over n >= 1 of a_n / n^s, where the a_n are given complex coefficients and s is a complex variable. Take a_n = 1 for all n and you recover zeta(s). Choose other coefficients — say a_n = (-1)^(n-1), or the values of a multiplicative arithmetic function, or the coefficients of a Dirichlet character — and you get a whole family of functions that package arithmetic information in analytic form. Where a power series sum a_n z^n is built from the building blocks z^n, a Dirichlet series is built from the building blocks 1/n^s = n^(-s).
The crucial feature is the geometry of convergence. A power series converges inside a DISK; a Dirichlet series converges inside a HALF-PLANE. There is a real number sigma_0, the abscissa of convergence, such that the series converges for every s with Re s > sigma_0 and diverges for Re s < sigma_0. On the dividing vertical line Re s = sigma_0 behaviour is delicate. Within the region of convergence the series defines a holomorphic function, and convergence is uniform on compact subsets, so you may differentiate term by term: the derivative is sum of -a_n (log n) / n^s. This half-plane picture is the Dirichlet-series analogue of the radius of convergence story for power series.
Dirichlet series are the language of analytic number theory. Encoding an arithmetic sequence a_n as a Dirichlet series turns questions about the average or distribution of a_n into questions about the analytic behaviour (poles, zeros, growth) of the resulting function — and complex analysis then answers them. The Euler product, when a_n is multiplicative, factors the series over primes just as it did for zeta. A caveat worth stating: unlike power series, a Dirichlet series can have a larger half-plane of ABSOLUTE convergence than of ordinary convergence; the two abscissae can differ (by at most 1), so 'converges' and 'converges absolutely' are genuinely different thresholds here.
The alternating series sum (-1)^(n-1) / n^s = 1 - 1/2^s + 1/3^s - ... is the Dirichlet eta function, and it converges for Re s > 0 (its abscissa of convergence is 0), a strictly bigger region than zeta's series. It relates to zeta by eta(s) = (1 - 2^(1-s)) zeta(s), which is one slick way to continue zeta into the strip 0 < Re s < 1.
Same n^(-s) skeleton, different coefficients: choosing a_n = (-1)^(n-1) widens the convergence half-plane to Re s > 0.
A Dirichlet series converges in a half-plane, not a disk — the analogue of a power series' radius of convergence is the abscissa of convergence. And do not assume convergence equals absolute convergence: the two abscissae can differ by up to 1.