Sums of singular series along arithmetic progressions and with smooth weights

TL;DR

This paper investigates sums of singular series related to prime distributions, focusing on arithmetic progressions and smooth weights, revealing how these sums are influenced by modular incidences and pairings of weights.

Contribution

It introduces a framework for analyzing constrained sums of singular series, connecting their values to modular incidences and pairings of smooth functions, advancing understanding of prime distribution patterns.

Findings

Sum values are governed by modular incidences in arithmetic progressions.

Pairings of smooth functions determine sum values with weights.

Results provide insights into singular series in various formats.

Abstract

Sums of the singular series constants that appear in the Hardy--Littlewood $k$-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo $r$ or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo $r$ of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats.