On additive convolution sum of arithmetic functions and related questions
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arXiv
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| Format: | Preprint |
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2025
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| author | Misra, Bikram Saha, Biswajyoti Sharma, Anubhav |
| author_facet | Misra, Bikram Saha, Biswajyoti Sharma, Anubhav |
| contents | Ingham studied two types of convolution sums of the divisor function, namely 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 work on this convolution sum, but only little has been done towards the additive convolution sum. In this article, we extend the classical result Ingham to derive an asymptotic formula with an error term of the sub-sum $\sum_{n < M} d(n) d(N-n)$ for an integer $M \le N$. Using this, we 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 turm leads us to a well-established prediction of Ramanujan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On additive convolution sum of arithmetic functions and related questions Misra, Bikram Saha, Biswajyoti Sharma, Anubhav Number Theory 11A25, 11N37 Ingham studied two types of convolution sums of the divisor function, namely 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 work on this convolution sum, but only little has been done towards the additive convolution sum. In this article, we extend the classical result Ingham to derive an asymptotic formula with an error term of the sub-sum $\sum_{n < M} d(n) d(N-n)$ for an integer $M \le N$. Using this, we 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 turm leads us to a well-established prediction of Ramanujan. |
| title | On additive convolution sum of arithmetic functions and related questions |
| topic | Number Theory 11A25, 11N37 |
| url | https://arxiv.org/abs/2502.08305 |