#### Date of Award

1-1-2021

#### Document Type

Dissertation

#### Degree Name

Ph.D. in Mathematics

#### First Advisor

Micah B. Milinovich

#### Second Advisor

Gerard J. Buskes

#### Third Advisor

Ayla Gafni

#### Relational Format

dissertation/thesis

#### Abstract

Number theorists have made great progress in understanding the distribution of the prime numbers by studying properties of the Riemann zeta-function. A celebrated and classical result of Selberg is that the real and imaginary parts of the logarithm of Riemann zeta-function are normally distributed on the critical line. Selberg proved this using the method of moments. It is known that any model of the logarithm of the Riemann zeta-function near the critical line requires input from the primes and the zeros of the zeta function. We refine Selberg's calculation for the variance of the real part of the logarithm of the Riemann zeta-function (the second moment) assuming the Riemann Hypothesis (RH) and carefully studying the pair correlation of the zeros. This uses ideas of Montgomery and Goldston and tools from Fourier analysis. Then we consider the distribution of real and imaginary parts of the logarithm of the Riemann zeta-function in short intervals, proving an asymptotic for the mean-square of the differences of shifted values uniformly in bounded intervals. Our results generalize previous work of Fujii and establish a conjecture of Berry for the number variance of the zeros, assuming RH, and a conjecture for the pair correlation of zeta zeros in long ranges.

#### Recommended Citation

Lugar, Meghann Moriah, "On the Variance in Selberg’s Central Limit Theorem Dissertation" (2021). *Electronic Theses and Dissertations*. 2160.

https://egrove.olemiss.edu/etd/2160

#### Concentration/Emphasis

Mathematics