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Chiara Bellotti

Publications and source records attributed to Chiara Bellotti.

12 recordsLinked to original sources
arXiv2026

Counting zeros of Artin $L$-functions

In this article, assuming Artin's (holomorphy) conjecture, we establish an explicit asymptotic formula for the number of non-trivial zeros, up to any given height $T\geq 1$, of Artin $L$-functions. As a consequence, our result yields an unconditional explicit zero-counting formula for Hecke $L$-functions over any number field. In addition, our result improves the recent work of Amberger on Dedekind and Riemann zeta functions and the previous work of Bennett-Martin-O'Bryant-Rechnitzer on Dirichlet $L$-functions for sufficiently large $T$.

math.NT↗
arXiv2026

Zero-free regions inspired by work of Heath-Brown

We prove a new explicit zero-free region for the Riemann zeta-function, drawing substantially on Heath-Brown's seminal work on Linnik's constant. Using these ideas we are able to prove that $\zeta(\sigma + it)\ne 0$ whenever $t\geq 3$ and $\sigma \geq 1- 1/(4.896\log t)$.

math.NT↗
arXiv2025

A new zero-density estimate for $\zeta(s)$ and the error term in the Prime Number Theorem

We will provide a new type of zero-density estimate for $\zeta(s)$ when $\sigma$ is sufficiently close to $1$. In particular, we will show that $N(\sigma,T)$ can be bounded by an absolute constant when $\sigma$ is sufficiently close to the left edge of the Korobov-Vinogradov zero-free region. As a consequence, we provide the optimal error term in the prime number theorem of the form $$ \psi(x)-x \ll x\exp \left\{-(1-\varepsilon) \omega(x)\right\},\qquad \omega(x):=\min _{t \geq 1}\{\nu(t) \log x+\log t\}, $$ where $\nu(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$ is a decreasing function such that $\zeta(\sigma+it)\neq 0$ for $\sigma\ge 1-\nu(t)$. Precisely, we will show that we can take $\varepsilon=0$.

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arXiv2024

Improved estimates for the argument and zero-counting function of the Riemann zeta-function

In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing \[ \left| N (T) - \frac{T}{ 2 \pi} \log \left( \frac{T}{2\pi e}\right) \right|\le 0.10076\log T+0.24460\log\log T+8.08344, \] for every $T\ge e$, where $N(T)$ is the number of non-trivial zeros $\rho=\beta+i\gamma$, with $0<\gamma \le T$, of the Riemann zeta-function $\zeta(s)$. The main source of improvement comes from implementing new subconvexity bounds for $\zeta(\sigma+it)$ on some $\sigma_k$-lines inside the critical strip.

math.NT↗
arXiv2024

An explicit log-free zero density estimate for the Riemann zeta-function

We will provide an explicit log-free zero-density estimate for $ζ(s)$ of the form $N(σ,T)\le AT^{B(1-σ)}$. In particular, this estimate becomes the sharpest known explicit zero-density estimate uniformly for $σ\in[α_0,1]$, with $0.985\le α_0\le 0.9927$ and $3\cdot 10^{12}<T\le \exp(6.7\cdot 10^{12})$.

math.NT↗
arXiv2023

Explicit zero density estimate near unity

We will provide the first explicit zero-density estimate for $ζ$ of the form $N(σ,T)\le \mathcal{C}T^{B(1-σ)^{3/2}}(\log T)^C$. In particular, we improve $C$ to $10393/900=11.547\dots.$

math.NT↗
arXiv2023

Explicit bounds for the Riemann zeta function and a new zero-free region

We prove that $|ζ(σ+it)|\le 70.7 |t|^{4.438 (1-σ)^{3/2}}\log^{2/3}|t|$ for $1/2\leσ\le 1$ and $|t|\ge 3$. As a consequence, we improve the explicit zero-free region for $ζ(s)$, showing that $ζ(σ+it)$ has no zeros in the region $σ\geq 1-1 /\left(54.004(\log |t|)^{2 / 3}(\log \log |t|)^{1 / 3}\right)$ for $|t| \geq 3$ and asymptotically in the region $σ\geq 1-1 /\left(48.0718(\log |t|)^{2 / 3}(\log \log |t|)^{1 / 3}\right)$ for $|t|$ sufficiently large.

math.NT↗
arXiv2023

On the generalised Dirichlet divisor problem

We improve unconditional estimates on $Δ_k(x)$, the remainder term of the generalised divisor function, for large $k$. In particular, we show that $Δ_k(x) \ll x^{1 - 1.889k^{-2/3}}$ for all sufficiently large fixed $k$.

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arXiv2022

New bounds for numbers of primes in element orders of finite groups

Let $ρ(n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group $G$, all elements of which have orders divisible by at most $n$ distinct primes. We show that $ρ(n)\leq 5n$ for all $n\geq 1$. As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.

math.GR↗
arXiv2022

Elementary methods in the study of Deuring-Heilbronn Phenomenon

The aim of this work is to improve some elementary results regarding both the Deuring-Phenomenon and the Heilbronn-Phenomenon. We will give better estimates regarding both the influence of zeros of the Riemann zeta function on the exceptional zeros and that of the non-trivial zeros of arbitrary L-functions belonging to non-principal characters on the exceptional zeros.

math.NT↗