A Dirichlet series is an infinite series of the form $\sum_{n=1}^\infty \frac{a_n}{n^s}$, where $a_n$ are complex coefficients and $s$ is a complex variable. These series have applications in number theory, complex analysis, and other areas of mathematics.
The Dirichlet series convergence theorem states that if a Dirichlet series converges at a point $s_0$ in the complex plane, then it converges absolutely at all points $s$ with $\mathrm{Re}(s) > \mathrm{Re}(s_0)$. This theorem is useful for determining the convergence of Dirichlet series.
The Riemann zeta function, denoted by $\zeta(s)$, is closely related to Dirichlet series. In fact, the Riemann zeta function can be expressed as a Dirichlet series: $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. This connection allows for the study of the Riemann zeta function using techniques from Dirichlet series theory.
The analytic continuation of a Dirichlet series is the process of extending the definition of the series to a larger region in the complex plane. This allows for the evaluation of the series at points where it may not have initially converged. Analytic continuation is a powerful tool for studying the behavior of Dirichlet series.
Dirichlet series have many applications in number theory. For example, the Riemann zeta function, which can be expressed as a Dirichlet series, is closely related to the distribution of prime numbers. Other Dirichlet series, such as the Dirichlet L-series, are also important in number theory, particularly in the study of quadratic forms and quadratic characters.