arXiv subjects

Bikram Misra

Publications and source records attributed to Bikram Misra.

3 recordsLinked to original sources

arXiv2025

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

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 $h \le x^{1-\epsilon}$ for any $\epsilon>0$ 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.

math.NT↗

arXiv2025

A triple convolution sum of the divisor function

We study the triple convolution sum of the divisor function given by $$\sum_{n\leq x} d(n)d(n-h)d(n+h)$$ for $h\neq 0$ and $d(n)$ denotes the number of positive divisors of $n$. Based on algebraic and geometric considerations, Browning conjectured that the above sum is asymptotic to $c_hx(\log x)^3$, for a suitable constant $c_h\neq 0$, as $x\to \infty$. This conjecture is still unproved. Using sieve-theoretic results of Wolke and Nair (respectively), it is possible to derive the exact order of the sum. The lower bound of the correct order of magnitude can also be derived by very elementary arguments. In this paper, using the Tauberian theory for multiple Dirichlet series, we prove an explicit lower bound and provide a new theoretical framework to predict Browning's conjectured constant $c_h$.

math.NT↗

arXiv2025

On additive convolution sum of arithmetic functions and related questions

Ingham studied two types of convolution sums of the divisor function, the shifted convolution sum $\sum_{n \le N} d(n) d(n+h)$ and the additive convolution sum $\sum_{n < N} d(n) d(N-n)$ for integers $N, h$ and derived their asymptotic formulas as $N \to \infty$. There have been numerous works extending Ingham's result on the shifted convolution sum, but only little has been done towards the additive convolution sum. In this article, we extend the classical result of Ingham to derive an asymptotic formula with an error term of the sub-sum $\sum_{n < M} d(n) d(N-n)$ for certain integers $M \le N$. This involves careful choice of an applicable range of $M$. We also study the convolution sum $\sum_{n < M} f(n) g(N-n)$ for certain arithmetic functions $f$ and $g$ with absolutely convergent Ramanujan expansions, which in turn leads us to a well-established prediction of Ramanujan.

math.NT↗