arXiv subjects
Publications and source records attributed to Brad Rodgers.
We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet $L$-functions and our proof relies on the asymptotic large sieve.
Let $t$ be random and uniformly distributed in the interval $[T,2T]$, and consider the quantity $N(t+1/\log T) - N(t)$, a count of zeros of the Riemann zeta function in a box of height $1/\log T$. Conditioned on the Riemann hypothesis, we show that the probability this count is greater than $x$ decays at least as quickly as $e^{-Cx\log x}$, uniformly in $T$. We also prove a similar results for the logarithmic derivative of the zeta function, and likewise analogous results for the eigenvalues of a random unitary matrix. We use results of this sort to show on the Riemann hypothesis that the averages $$ \frac{1}{T} \int_T^{2T} \Bigg| \frac{ζ\Big(\frac{1}{2} + \fracα{\log T} + it\Big)}{ζ\Big(\frac{1}{2}+ \fracβ{\log T} + it\Big)}\Bigg|^m\,dt $$ remain bounded as $T\rightarrow\infty$, for $α, β$ complex numbers with $β\neq 0$. Moreover we show rigorously that the local distribution of zeros asymptotically controls ratio averages like the above; that is, the GUE Conjecture implies a (first-order) ratio conjecture.
We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have been proved under the assumption of the Riemann hypothesis. The result mirrors central limit theorems in random matrix theory that have been proved by Szegő, Spohn, and Soshnikov among others, and therefore provides support for the view that the zeros of the zeta function are distributed like the eigenvalues of a random matrix. A key ingredient in our proof is a simple bootstrapping of classical zero density estimates of Selberg and Jutila for the zeta function, which may be of independent interest.
We study the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements. In the limit as $q\rightarrow\infty$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $d_k(n)$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of certain polynomials, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.
We estimate the covariance in counts of almost-primes in $\mathbb{F}_q[T]$, weighted by higher-order von Mangoldt functions. The answer takes a pleasant algebraic form. This generalizes recent work of Keating and Rudnick that estimates the variance of primes, and makes use, as theirs, of a recent equidistribution result of Katz. In an appendix we prove some related identities for random matrix statistics, which allows us to give a quick proof of a $2\times2$ ratio theorem for the characteristic polynomial of the unitary group. We additionally identify arithmetic functions whose statistics mimic those of hook Schur functions.
On the assumption of the Riemann hypothesis, we show that over a class of sufficiently smooth test functions, a measure conjectured by Bogomolny and Keating coincides to a very small error with the actual pair correlation measure for zeroes of the Riemann zeta function. Our result extends the well known result of Montgomery that over the same class of test functions the pair correlation measure coincides (to a larger error term) with that of the Gaussian Unitary Ensemble (GUE). The restriction of test functions remains stringent, but we are nonetheless able to detect, at a microscopically blurred resolution, macroscopic troughs in the pair correlation measure.
On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.