Short Exponential Sums and Ternary Correlations of Multiplicative Functions

TL;DR

This paper studies the average behavior of ternary correlations among multiplicative functions using novel exponential sum bounds and integral moment estimates for $L$-functions, diverging from traditional spectral methods.

Contribution

It introduces a new approach combining the circle method with short exponential-sum bounds based on $L$-function moments, advancing the analysis of multiplicative functions.

Findings

Derived new short exponential-sum estimates from $L$-function moment bounds.

Established average bounds for ternary correlations of multiplicative functions.

Provided a framework that bypasses spectral theory and Heath-Brown decompositions.

Abstract

In this paper, we investigate the average behavior of ternary correlations for general $k$-divisor-bounded multiplicative functions, assuming certain second moment integral bounds for the associated $L$-functions. Our approach differs from previous methods based on spectral theory or Heath-Brown-type decompositions, and instead combines the circle method with weighted short exponential-sum bounds. The key input is short exponential-sum estimates obtained from integral moment bounds for $L$-functions.