arXiv subjects

Barnabás Szabó

Publications and source records attributed to Barnabás Szabó.

3 recordsLinked to original sources

arXiv2024

A lower bound on high moments of character sums

For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{ϕ(q)} \sum_{\substack{χ\text{ mod }q\\ χ\neq χ_0}} \Big| \sum_{n\leq x} χ(n)\Big|^{2k}, \end{equation*} where $1\leq x\leq q$, and $χ$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method.

math.NT↗

arXiv2023

High moments of theta functions and character sums

Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet $L$-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen theorems of Munsch, who proved almost sharp upper bounds for these quantities. The main new ingredient of our proof comes from a paper of Harper, who showed the related result $\int_{0}^T |ζ(1/2+it)|^{2k} \ll_k T(\log T)^{k^2} $ for all $k\geq 0$ under the Riemann Hypothesis. Finally, we obtain a sharp conditional upper bound on high moments of character sums of arbitrary length.

math.NT↗

arXiv2022

On the existence of products of primes in arithmetic progressions

We study the existence of products of primes in arithmetic progressions, building on the work of Ramaré and Walker. One of our main results is that if $q$ is a large modulus, then any invertible residue class mod $q$ contains a product of three primes where each prime is at most $q^{6/5+ε}$. Our arguments use results from a wide range of areas, such as sieve theory or additive combinatorics, and one of our key ingredients, which has not been used in this setting before, is a result by Heath-Brown on character sums over primes from his paper on Linnik's theorem.

math.NT↗