Editing Analytic number theory (section)

=== Dirichlet series ===
{{main|Dirichlet series}}
One of the most useful tools in multiplicative number theory are [[Dirichlet series]], which are functions of a complex variable defined by an infinite series of the form

:<math>f(s)=\sum_{n=1}^\infty a_nn^{-s}.</math>

Depending on the choice of coefficients <math>a_n</math>, this series may converge everywhere, nowhere, or on some half plane. In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane. The utility of functions like this in multiplicative problems can be seen in the formal identity

:<math>\left(\sum_{n=1}^\infty a_nn^{-s}\right)\left(\sum_{n=1}^\infty b_nn^{-s}\right)=\sum_{n=1}^\infty\left(\sum_{k\ell=n}a_kb_\ell\right)n^{-s};</math>

hence the coefficients of the product of two Dirichlet series are the [[multiplicative convolution]]s of the original coefficients. Furthermore, techniques such as [[partial summation]] and [[Tauberian theorem]]s can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series (or a product of simpler Dirichlet series using convolution identities), examine this series as a complex function and then convert this analytic information back into information about the original function.