arXiv subjects
Publications and source records attributed to Bryce Kerr.
Let $S_χ(N)= \sum_{n\leq N} χ(n)$, where $χ$ is a Dirichlet character modulo a prime $p$. For integral $r\geq 2$, Burgess gave bounds of the shape $S_χ(N) \ll N^{1-1/r} p^{(r+1)/(4r^{2})} \log p$. The factor of $\log p$ was reduced by several authors. We show that it can be removed completely.
We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free palindromes in base $b$. We also provide an asymptotic expression for the number of such integers $\leq x$. The core of our proof utilises a hybrid $p$-adic/Archimedean van der Corput process, used in conjunction with an equidistribution estimate of Tuxanidy and Panario, as well as an elementary argument of Cilleruelo, Luca and Shparlinski.
In this paper we consider the error term in the expected count of squarefree values of a polynomial when evaluated at integers \(n\le N\). We prove a power saving on average over polynomials of fixed degree and height at most \(H\). Our results improve previous estimates of Browning and Shparlinski in that the required range of \(H\) is independent of the degree, but depends on \(N\)
In this article we improve Heath-Brown's estimate for the error term in the three-dimensional sphere problem, proving \[ \#\{\mathbf x\in\mathbb Z^3:\|\mathbf x\|\le R\}=\frac{4π}{3}R^3+O_\varepsilon(R^{55/42+\varepsilon}). \] Incoporating the recent OpenAI quasi-GRH into our argument further improves this to \[ \#\{\mathbf x\in\mathbb Z^3:\|\mathbf x\|\le R\}= \frac{4π}{3}R^3+O_\varepsilon(R^{\frac{2274811}{1743872}+\varepsilon}). \]
Let $P$ be a trigonometric polynomial with only real zeros. We prove that at most $\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $ζ(s)$ on the critical line. We also show that \begin{equation*} \sup_{\substack{T\leq x\leq2T\\ P(x)=0}} \log \left|ζ\left(\frac{1}{2}+ix\right)\right| \geq (1+o(1))\sqrt{\frac{\log T}{6\log\log T}}. \end{equation*} These results extend the corresponding theorems of Li and Radziwill on the zeros of $ζ(s)$ along vertical arithmetic progressions. The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\sum_{P(γ)=0} {\bf δ}_γ$ is crystalline.
This paper simplifies the recent OpenAI proof that there are no Landau--Siegel zeros.
We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.
We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,χ)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $χ$ runs through the ratio set ${\mathcal A}/{\mathcal A}$ of a thin set ${\mathcal A}$ of characters with small product set modulo a prime $q$. Some of these bounds are based on new zero-density estimates on average over sets of characters with a small product set. These bounds follow from a mean value estimate for character sums, which is based on the work of D. R. Heath-Brown (1979). As yet another application of this mean value estimate, we obtain an unconditional version of a conditional (on the Generalised Riemann Hypothesis) result of Z. Rudnick and A. Zaharescu (2000) about gaps between primitive roots.
It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and computational techniques. In this paper, we establish new results concerning explicit bounds for cubic non-residues and refine previous computational techniques, enabling us to completely characterise all norm-Euclidean cyclic cubic fields.
We establish new conditions under which a sequence of real numbers has metric Poissonian pair correlation. These conditions strengthen results of Aistleitner, El-Baz and Munsch (2021) and resolve one of their open problems under a mild growth assumption. As applications, we show that quantitatively convex and polynomial sequences have metric Poissonian pair correlation.
Let $L(s,π)=\sum_{n=1}^{\infty}λ_π(n)n^{-s}$ be an $L$-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $λ_π(n)$ strongly repel the low-lying zeros of $L(s,π)$ away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.
We obtain bounds on the average size of Bohr sets with coefficients parametrised by polynomials over finite fields and obtain a series of general results and also some sharper results for specific sets which are important for applications to computer science. In particular, we use our estimates to show that a heuristic assumption used in the many variable version of Coppersmith's method holds with high probability. We demonstrate the use of our results on the approximate greatest common divisor problem and obtain a fully rigorous version of the heuristic algorithm of H. Cohn and N. Heninger (2013).
We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet $L$-functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.
In recent decades, the use of ideas from Minkowski's Geometry of Numbers has gained recognition as a helpful tool in bounding the number of solutions to modular congruences with variables from short intervals. In 1941, Mahler introduced an analogue to the Geometry of Numbers in function fields over finite fields. Here, we build on Mahler's ideas and develop results useful for bounding the sizes of intersections of lattices and convex bodies in $\mathbb{F}_q((1/T))^d$, which are more precise than what is known over $\mathbb{R}^d$. These results are then applied to various problems regarding bounding the number of solutions to congruences in $\mathbb{F}_q[T]$, such as the number of points on polynomial curves in low dimensional subspaces of finite fields. Our results improve on a number of previous bounds due to Bagshaw, Cilleruelo, Shparlinski and Zumalacárregui. We also present previous techniques developed by various authors for estimating certain energy/point counts in a unified manner.
In this paper we generalize a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefcients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial congruences and polynomial values and obtain some new results for mixed character sums.
Assume the Riemann hypothesis throughout. We obtain some new estimates for the size of the set of large values of the error term in the prime number theorem. Our argument is based on an analysis of the behavior of zeros of the Riemann zeta function in Bohr sets.
Turán observed that logarithmic partial sums $\sum_{n\le x}\frac{f(n)}{n}$ of completely multiplicative functions (in the particular case of the Liouville function $f(n)=λ(n)$) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. Firstly, we show that for every $\varepsilon>0$ there exists some $x_0\ge 1,$ such that for any completely multiplicative function $f$ satisfying $-1\le f(n)\le 1$, we have $$\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{1}{(\log\log{x})^{1-\varepsilon}}, \quad x\ge x_0.$$ This improves a previous bound due to Granville and Soundararajan. Secondly, we show that if $f$ is a typical (random) completely multiplicative function $f:\mathbb{N}\to \{-1,1\}$, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is negative for a given large $x,$ is $O(\exp(-\exp(\frac{\log x\cdot \log\log\log x}{C\log \log x}))).$ This improves on recent work of Angelo and Xu.
We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products and reciprocals of linear forms. This allows us to make some progress towards a question of B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev and I. D. Shkredov (2019) on an extreme case of the Erdős-Szemerédi conjecture in finite fields.