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Triple convolution sums of the generalised divisor functions and related sums over primes

Bikram Misra, Biswajyoti Saha

TL;DR

This work analyzes the triple convolution sums of generalized divisor functions, proving a predicted main-term asymptotic with an explicit, multiplicative constant $\nabla_{k,l,m}(h)$ when $h\le x^{1-\varepsilon}$, and establishing a matching lower bound via Tauberian theory for Dirichlet series in several complex variables. It extends the analysis to the prime-sum variant, $\mathcal{T}'(d_k,d_l,d_m;x,h)$, deriving explicit lower bounds by combining multi-variable Tauberian results with Bombieri–Vinogradov-type bounds. The authors compute the main constant $\nabla_{k,l,m}(h)$ as an Euler product, and provide a probabilistic interpretation that explains its structure through local prime factors. The techniques blend multi-variable Dirichlet series, sharp divisor-function identities, and sieve-type input to yield concrete lower bounds and a deeper understanding of the arithmetic structure governing these higher-order convolution sums.

Abstract

We study the triple convolution sum of the generalised divisor functions $$\sum_{n\leq x} d_k(n+h)d_l(n)d_m(n-h),$$ where $h \le x^{1-ε}$ for any $ε>0$ and $d_k(n)$ denotes the generalised divisor function which counts the number of ways $n$ can be written as a product of $k$ many positive integers. The purpose of this paper is three-fold. Firstly, we note a predicted asymptotic estimate for the above sum, where the constant appearing in the estimate can be obtained from the theory of Dirichlet series of several complex variables and also using some probabilistic arguments. Then we show that a lower bound of the correct order can be derived using the several variable Tauberian theorems, where, more importantly, the constant in the predicted asymptotic can be recovered. Lastly, in the spirit of the Titchmarsh divisor problem, we consider this triple convolution sum over the prime numbers, which essentially leads to a shifted convolution sum. We use the Tauberian theory of multiple Dirichlet series along with the Bombieri-Vinogradov theorem to derive an explicit lower bound of this.

Triple convolution sums of the generalised divisor functions and related sums over primes

TL;DR

This work analyzes the triple convolution sums of generalized divisor functions, proving a predicted main-term asymptotic with an explicit, multiplicative constant when , and establishing a matching lower bound via Tauberian theory for Dirichlet series in several complex variables. It extends the analysis to the prime-sum variant, , deriving explicit lower bounds by combining multi-variable Tauberian results with Bombieri–Vinogradov-type bounds. The authors compute the main constant as an Euler product, and provide a probabilistic interpretation that explains its structure through local prime factors. The techniques blend multi-variable Dirichlet series, sharp divisor-function identities, and sieve-type input to yield concrete lower bounds and a deeper understanding of the arithmetic structure governing these higher-order convolution sums.

Abstract

We study the triple convolution sum of the generalised divisor functions where for any and denotes the generalised divisor function which counts the number of ways can be written as a product of many positive integers. The purpose of this paper is three-fold. Firstly, we note a predicted asymptotic estimate for the above sum, where the constant appearing in the estimate can be obtained from the theory of Dirichlet series of several complex variables and also using some probabilistic arguments. Then we show that a lower bound of the correct order can be derived using the several variable Tauberian theorems, where, more importantly, the constant in the predicted asymptotic can be recovered. Lastly, in the spirit of the Titchmarsh divisor problem, we consider this triple convolution sum over the prime numbers, which essentially leads to a shifted convolution sum. We use the Tauberian theory of multiple Dirichlet series along with the Bombieri-Vinogradov theorem to derive an explicit lower bound of this.

Paper Structure

This paper contains 15 sections, 10 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary facts and key results related to the generalised divisor functions
  4. Results from the multiple Dirichlet series theory
  5. Bombieri-Vinogradov theorem
  6. Proof of Theorem \ref{['lower bound of triple']}
  7. Estimating $S_1(x;h)$
  8. Proof of Theorem \ref{['three-div-function']}
  9. Proof of Theorem \ref{['lower bound of triple sum over primes']}
  10. Estimating $S_1'(x;h)$
  11. Proof of Theorem \ref{['three-div-function-prime']}
  12. Proof of Remark \ref{['rmk']}
  13. Getting to $\nabla_{k,l,m}(h)$ using probabilistic route
  14. Proof of Lemma \ref{['Expectations calculation']}
  15. Verifying $C_{k,l,m}(h)=\psi_{k,l,m}(h)$

Key Result

Theorem 1.1

Let $\epsilon>0$ and integers $k,l,m\ge 2$. Then for $h\le x^{1-\epsilon}$, as $x \to \infty$.

Theorems & Definitions (16)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 6 more