Work

Harper’s Beyond Square-Root Conjecture

Year: 2025

Type: article

Abstract: We explain how the (shifted) Ratios Conjecture for $L(s,\chi )$ would extend a randomization argument of Harper from a conductor-limited range to an unlimited range of “beyond square-root cancellation” for character twists of the Liouville function. As a corollary, the Liouville function would have nontrivial cancellation in arithmetic progressions of modulus just exceeding the well-known square-root barrier. Morally, the paper passes from random matrices to random multiplicative functions.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf279

Authors: Victor Y Wang, Max Wenqiang Xu

Institutions: Institute of Science and Technology Austria, Courant Institute of Mathematical Sciences, Institute of Mathematics, Academia Sinica, Academia Sinica

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W4414090271

FWCI: 3.445

Cites: 34

Cited by: 1

Related to: 10