Year: 2026
Type: preprint
Abstract: We study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for $x\geqslant21$ we have \begin{multline*} |ψ(x)-x| < 0.239\, x \exp\bigl(-0.1982767\,(1+\log\log\log(x)/(15\log\log(x)))\log(x)^{3/5}\log\log(x)^{-1/5}\bigr). \end{multline*} Combined with the zero-free half-plane $\Re(s)>7/8$ announced by OpenAI in October 2026, the same methods give $|ψ(x)-x| < 0.04934\,x^{3/4}\log(x)^2 + 0.02636425\,x^{7/8}$ for $x\geqslant 21$.
Source: arXiv (Cornell University)
Authors: Fiori, Andrew, Jaskari, Mikko
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G2271590008
OpenAlex ID: W7214051944