A Dirichlet series is a type of infinite series that takes the form \sum_n=1^\infty \fraca_nn^s , where a_n are complex coefficients and s is a complex variable. These series are often used in number theory and have applications in studying the distribution of prime numbers. One of the most famous examples of a Dirichlet series is the Riemann zeta function, denoted as \zeta(s) , which is defined for s > 1 as \sum_n=1^\infty \frac1n^s . Dirichlet series can converge for certain values of s and are instrumental in analytic number theory, particularly in understanding properties of prime numbers.