Improved Averaged Distribution of $d_3(n)$ in Prime Arithmetic Progressions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908985302450176 |
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| author | Aydemir, Metin Can Boran, Muhammet |
| author_facet | Aydemir, Metin Can Boran, Muhammet |
| contents | We say that $d_3(n)$ has exponent of distribution $θ$ if, for every $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{θ-\varepsilon}$. Nguyen proved, following earlier work of Banks, Heath-Brown, and Shparlinski, that after averaging over reduced residue classes $a \bmod q$, the function $d_3(n)$ has exponent of distribution $2/3$. Using the Petrow--Young subconvexity bound for Dirichlet $L$-functions, we improve this to $8/11$ when averaging over residue classes modulo a prime $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improved Averaged Distribution of $d_3(n)$ in Prime Arithmetic Progressions Aydemir, Metin Can Boran, Muhammet Number Theory Primary 11N37, Secondary 11M06 We say that $d_3(n)$ has exponent of distribution $θ$ if, for every $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{θ-\varepsilon}$. Nguyen proved, following earlier work of Banks, Heath-Brown, and Shparlinski, that after averaging over reduced residue classes $a \bmod q$, the function $d_3(n)$ has exponent of distribution $2/3$. Using the Petrow--Young subconvexity bound for Dirichlet $L$-functions, we improve this to $8/11$ when averaging over residue classes modulo a prime $q$. |
| title | Improved Averaged Distribution of $d_3(n)$ in Prime Arithmetic Progressions |
| topic | Number Theory Primary 11N37, Secondary 11M06 |
| url | https://arxiv.org/abs/2601.12601 |