New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach

What's new 2025-01-29

Timothy Trudgian, Andrew Yang and I have just uploaded to the arXiv the paper “New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach“. This paper launches a project envisioned in this previous blog post, in which the (widely dispersed) literature on various exponents in classical analytic number theory, as well as the relationships between these exponents, are collected in a living database, together with computer code to optimize the relations between them, with one eventual goal being to automate as much as possible the “routine” components of many analytic number theory papers, in which progress on one type of exponent is converted via standard arguments to progress on other exponents.

The database we are launching concurrently with this paper is called the Analytic Number Theory Exponent Database (ANTEDB). This Github repository aims to collect the latest results and relations on exponents such as the following:

The growth exponent ${\mu(\sigma)}$ of the Riemann zeta function ${\zeta(\sigma+it)}$ at real part ${\sigma}$ (i.e., the best exponent for which ${\zeta(\sigma+it) \ll |t|^{\mu(\sigma)+o(1)}}$ as ${t \rightarrow \infty}$ );
Exponent pairs ${(k,\ell)}$ (used to bound exponential sums ${\sum_{n \in I} e(T F(n/N))}$ for various phase functions ${F}$ and parameters ${T,N}$ );
Zero density exponents ${A(\sigma)}$ (used to bound the number of zeros of ${\zeta}$ of real part larger than ${\sigma}$ );

etc.. These sorts of exponents are related to many topics in analytic number theory; for instance, the Lindelof hypothesis is equivalent to the assertion ${\mu(1/2)=0}$ .

Information on these exponents is collected both in a LaTeX “blueprint” that is available as a human-readable set of web pages, and as part of our Python codebase. In the future one could also imagine the data being collected in a Lean formalization, but at present the database only contains a placeholder Lean folder.

As a consequence of collecting all the known bounds in the literature on these sorts of exponents, as well as abstracting out various relations between these exponents that were implicit in many papers in this subject, we were then able to run computer-assisted searches to improve some of the state of the art on these exponents in a largely automated fashion (without introducing any substantial new inputs from analytic number theory). In particular, we obtained:

four new exponent pairs;
several new zero density estimates; and
new estimates on the additive energy of zeroes of the Riemann zeta function.

We are hoping that the ANTEDB will receive more contributions in the future, for instance expanding to other types of exponents, or to update the database as new results are obtained (or old ones added). In the longer term one could also imagine integrating the ANTEDB with other tools, such as Lean or AI systems, but for now we have focused primarily on collecting the data and optimizing the relations between the exponents.