# An explicit formula for Dirichlet's L-Function

Ledoan, Andrew

## Department Examiner

Gao, Cuilan; Nichols, Roger

## Department

Dept. of Mathematics

## Publisher

University of Tennessee at Chattanooga

## Place of Publication

Chattanooga (Tenn.)

## Abstract

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 of Landau’s explicit formula that is uniform in both variables x and T. This explicit formula was used to estimate various sums involving the zeros of the zeta function, such as the discrete mean value formula for the zeta function. The purpose of this thesis is to obtain a generalization of Landau’s and Gonek’s explicit formulas in terms of the zeros of the Dirichlet L-function. To accomplish this, we employ the argument principal, Cauchy’s residue theorem, and an inequality of Selberg.

## Acknowledgments

First and foremost, I would like to thank my thesis advisor Dr. Andrew Ledoan of The University of Tennessee at Chattanooga. Without his assistance and dedicated involvement in every step of the process, this thesis would have never been accom- plished. I would like to thank him for his constant support and compassion over the past year. I would also like to express my gratitude to the members of my examina- tion committee, Dr. Roger Nichols and Dr. Cuilan Gao, for reading various drafts of my thesis. I would like to also express my appreciation for my high school math teacher, Mrs. Lesley Avery. She was the one who first showed me that mathemat- ics is beautiful, that mathematics is interesting, and that mathematics is useful. I would not be where I am today without her. Thank you to my friends for providing me with unfailing support and continuous encouragement throughout my study and through the process of writing this thesis. Finally, I would like to express my very profound gratitude to my parents Jagada and Michele. They showed me true love and support in its rarest form and how it can be used to overcome life’s challenges. My accomplishments would not have been possible without any of you. Thank you.

## Degree

B. S.; An honors thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Bachelor of Science.

5-2018

## Subject

Dirichlet principle; L-functions

## Keyword

Explicit formula; Analytic number theory; Prime numbers; Primes; Zeta-function; Zeta function

Mathematics

Theses

19 leaves

Text

English

## Rights

http://rightsstatements.org/vocab/InC/1.0/

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