On Additive Divisor Sums and minorants of divisor functions

2019/03/04 by Kevin M. Smith, Julio Andrade, Smith, Kevin +1
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Theories #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.1903.01566

Abstract

We establish asymptotic formulae for various correlations involving general divisor functions d_k(n) and partial divisor functions d_l(n,A)=∑_{q|n:q≤ n^A}d_l-1(q), where A∈[0,1] is a parameter and k,l∈ℕ are fixed. Our results relate the parameter A to the lengths of arithmetic progressions in which d_k(n) is uniformly distributed. As applications to additive divisor sums, we establish new lower bounds and a new equivalent condition for the conjectured asymptotic. We also prove a Tauberian theorem for general additive divisor sums.

Related

• On the Restricted Divisor Function in Arithmetic Progressions
• Local divisor correlations in almost all short intervals
• Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges
• On correlation of the 3-fold divisor function with itself
• On the general additive divisor problem