Physical Sciences and Mathematics Commons^™

An Explicit Formula For Dirichlet's L-Function, Shannon Michele Hyder

Honors Theses

The Riemann zeta function has a deep connection to the distribution of primes. In 1911 Landau proved that, an explicit formula where ρ = β + iγ denotes a complex zero of the zeta function and Λ(x) is an extension of the usual von Mangoldt function, so that Λ(x) = log p if x is a positive integral power of a prime p and Λ(x) = 0 for all other real values of x. Landau’s remarkable explicit formula lacks uniformity in x and therefore has limited applications to the theory of the zeta function. In 1993 Gonek proved a version …