Sign changes of the error term in the Piltz divisor problem

We study the function \Delta_k(x):=\sum_{n\leq x} d_k(n) - \mbox{Res}_{s=1} ( \zeta^k(s) x^s/s ), where $k\geq 3$ is an integer, $d_k(n)$ is the $k$-fold divisor function, and $\zeta(s)$ is the Riemann zeta-function. For a large parameter $X$, we show that there exist at least $X^{\frac{37}{96}-\varepsilon}$ disjoint subintervals of $[X,2X]$, each of length $X^{\frac{1}{2}-\varepsilon}$, such that $|\Delta_3(x)|\gg x^{1/3}$ for all $x$ in the subinterval. We show further that if the Lindel\"{o}f hypothesis is true and $k\geq 3$, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon}$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon}$, such that $|\Delta_k(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. If the Riemann hypothesis is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}} (\log X)^{-k^2-2}$. These results may be viewed as higher-degree analogues of a theorem of Heath-Brown and Tsang, who studied the case $k=2$. The first main ingredient of our proofs is a bound for the second moment of $\Delta_k(x+h)-\Delta_k(x)$. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of $\Delta_k(x)$, which we obtain by combining a method of Tsang with a technique of Lester.Comment: 28 page

Low-lying zeros of a large orthogonal family of automorphic LL-functions

We study a new orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q \asymp Q$. To illustrate our methods, we prove a one level density result for this family with the support of the Fourier transform of the test function being extended to be inside $(-4, 4)$. The main techniques developed in this paper will be useful in developing further results for this family, including estimates for high moments, information on the vertical distribution of zeros, as well as critical line theorems.Comment: 41 pages; minor revisio

Moments of zeta and correlations of divisor-sums: stratification and Vandermonde integrals

We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.Comment: 26 pages, clarified parts of the introduction, results unchange

On the zeros of Riemann's zeta-function

Thesis (Ph. D.)--University of Rochester. Department of Mathematics, 2017.This thesis has three parts. In the first part, we combine the mollifier method with a zero detection method of Atkinson to prove in a new way that a positive proportion of the nontrivial zeros of the Riemann zeta-function ζ(s) are on the critical line. One of the main ingredients of the proof is an estimate for a mollified fourth moment of ζ(1/2 + it). We deduce this estimate from the twisted fourth moment formula that has been recently developed by Hughes and Young. The second part of this thesis is concerned with bounding the number N(σ, T) of zeros of ζ(s) that have real parts > σ and imaginary parts between 0 and T. We prove a claim of Conrey that improves previous bounds for N(σ, T) due to Selberg and Jutila. We do this by constructing a new mollifier that allows us to apply Conrey’s technique of using Kloosterman sum estimates to deduce an asymptotic formula for the mollified second moment of ζ(σ + it) when σ is > 1/2 and close to 1/2 . In the third part of the thesis, we give a conditional proof of the equivalence of certain asymptotic formulas for (a) averages over intervals for the 2-point form factor F([character did not render], T) for the zeros of ζ(s), (b) the mean square of the logarithmic derivative of ζ(s), (c) a variance for the number of primes in short intervals, and (d) the number of pairs of zeros of ζ(s) with small gaps. The main result is a generalization of the fusion of a theorem of Goldston and a theorem of Goldston, Gonek, and Montgomery. We apply our result to deduce several consequences of the Alternative Hypothesis