# The simple zeros of the Riemann zeta-function

Ledoan, Andrew

## Department Examiner

Nichols, Roger; Gunasekera, Sumith; Chatzimanolis, Stylianos

## Department

Dept. of Mathematics

## Publisher

University of Tennessee at Chattanooga

## Place of Publication

Chattanooga (Tenn.)

## Abstract

There have been many tables of primes produced since antiquity. In 348 BC Plato studied the divisors of the number 5040. In 1202 Fibonacci gave an example with a list of prime numbers up to 100. By the 1770's a table of number factorizations up to two million was constructed. In 1859 Riemann demonstrated that the key to the deeper understanding of the distribution of prime numbers lies in the study of a certain complex-valued function, called the zeta-function. In 1973 Montgomery used explicit formulas to study the pair correlation of the zeros of the zeta-function and their relationship to primes. It is conjectured that all the zeros of the zeta-function are simple. Montgomery proved that at least two-thirds of the zeros are simple. In this thesis I provide complete proofs of Montgomery's method and its applications to simple zeros and differences between consecutive primes. In addition, I give a proof of the explicit formula derived by Ledoan and Zaharescu for the pair correlation of vertical shifts of zeros of the zeta-function and derive several consequences that may be useful for further study of the zeros.

## Acknowledgments

I wish to thank my advisor, Professor Andrew Ledoan, for his help and guidance throughout my departmental thesis. I would like to also thank my committee members, Professors Roger Nichols, Sumith Gunasekera, and Stylianos Chatzimanolis, for their time and support. Finally, I would like to thank my mother for her continuous support, my father for his unconditional love, and my sister for her friendship.

## 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.

5-2016

## Subject

Functions, Zeta; Riemann hypothesis

## Keyword

Riemann; Zeta-function; Montgomery; Prime numbers

Mathematics

Theses

iv, 39 leaves

Text

English

## Rights

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

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