Sums of singular series along arithmetic progressions and with smooth weights

Abstract

Sums of the singular series constants that appear in the Hardy–Littlewood [Formula: see text]-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 [Formula: see text] or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo [Formula: see text] 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.