New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach
Terence Tao, Tim Trudgian, Andrew Yang
TL;DR
The paper introduces the Analytic Number Theory Exponent Database (ANTEDB), a programmable framework that abstracts and systematises exponents arising in analytic number theory, enabling machine-assisted optimisation across interdependent bounds. Using ANTEDB, the authors derive four new exponent pairs, multiple new zero-density bounds for the Riemann zeta-function, and new estimates for the additive energy of zeta zeros, showcasing how automated reasoning over a network of known results can yield concrete improvements without new analytic input. The work also develops a comprehensive notation and machinery (e.g., LV, LVζ, and A-type exponents) and demonstrates how large-value theorems translate into sharp density and energy bounds, including sophisticated, piecewise-optimal results. By formalising results as Hypotheses in a Python-based system and computing optimal intersections of their implications, the paper provides a reproducible, extensible path toward sharper bounds in zeta-function theory and related L-functions, with potential extensions to log-free estimates and formal verification.
Abstract
We obtain several new bounds on exponents of interest in analytic number theory, including four new exponent pairs, new zero density estimates for the Riemann zeta-function, and new estimates for the additive energy of zeroes of the Riemann zeta-function. These results were obtained by creating the Analytic Number Theory Exponent Database (ANTEDB) to collect results and relationships between these exponents, and then systematically optimising these relationships to obtain the new bounds. We welcome further contributions to the database, which aims to allow easy conversion of new bounds on these exponents into optimised bounds on other related exponents of interest.