Dirichlet Series

where the an are complex numbers and s is a complex variable. Such functions are called Dirichlet series. We call a1 the constant term. A Dirichlet series will often be written as ∑ ann −s, with the index of summation understood to start at n = 1. Similarly, ∑ app −s runs over the primes, and ∑ apkp −ks runs over the prime powers excluding 1. (Not counting 1 as a prime power in that notation is reasonable in light of the way Dirichlet series that run over prime powers arise in practice, without a constant term.) A Dirichlet series over the prime powers that excludes the primes will be written ∑ p,k≥2 apkp −ks. The use of s as the variable in a Dirichlet series goes back to Dirichlet, who took s to be real and positive. Riemann emphasized the importance of letting s be complex. The convention of using σ and t for the real and imaginary parts of s seems to have become common at the beginning of the 20th century, and was universally adopted through the influence of Landau’s Handbuch [6] (1909).