A triple convolution sum of the divisor function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911131593867264 |
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| author | Misra, Bikram Murty, M. Ram Saha, Biswajyoti |
| author_facet | Misra, Bikram Murty, M. Ram Saha, Biswajyoti |
| contents | 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 $c_h$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_13082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A triple convolution sum of the divisor function Misra, Bikram Murty, M. Ram Saha, Biswajyoti Number Theory 11N37, 11M32, 11M45 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 $c_h$. |
| title | A triple convolution sum of the divisor function |
| topic | Number Theory 11N37, 11M32, 11M45 |
| url | https://arxiv.org/abs/2508.13082 |