{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-correlation-of-the-3-fold-divisor-function","title":"On correlation of the 3-fold divisor function with itself","arxiv_id":"2206.05877","date":"2022-06-13","proceeding":null,"authors":["David T. Nguyen"],"abstract":"Let $\\zeta^k(s) = \\sum_{n=1}^\\infty \\tau_k(n) n^{-s}, \\Re s > 1$. We present three conditional results on the ternary additive correlation sum $$\\sum_{n\\le X} \\tau_3(n) \\tau_3(n+h),\\quad (h\\ge 1),$$ and give numerical verifications of our method. The first is a conditional proof for the full main term of the above correlation sum for any composite shift $1 \\le h \\le X^{2/3}$, on assuming an averaged level of distribution for the three-fold divisor function $\\tau_3(n)$ in arithmetic progressions to level two-thirds. The second is a conditional derivation for the leading order main term asymptotics of this correlation sum, also valid for any composite shift $1 \\le h \\le X^{2/3}$. The third result gives a complete expansion of the polynomial for the full main term for the special case $h=1$ from both our method and from the delta-method, showing that our answers match. Our method is essentially elementary, especially for the $h=1$ case, uses congruences, and, as alluded to earlier, gives the same answer as in prior prediction of Conrey and Gonek [Duke Math. J. 107 (3) 2002], previously computed by Ng and Thom [Funct. Approx. Comment. Math. 60(1) 2019], and unpublished heuristic probabilistic arguments of Tao. Our procedure is general and works to give the full main term with a power-saving error term for any correlations of the form $\\sum_{n\\le X} \\tau_k(n) f(n+h)$, to any composite shift $h$, and for a wide class of arithmetic function $f(n)$.","url_abs":"https://arxiv.org/abs/2206.05877v2","url_pdf":"https://arxiv.org/pdf/2206.05877v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-correlation-of-the-3-fold-divisor-function","repo_url":"https://github.com/nguyen-d-8/correlations","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}