{"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/analytic-number-theory-and-algebraic","title":"Analytic Number Theory and Algebraic Asymptotic Analysis","arxiv_id":"2407.17820","date":"2024-07-25","proceeding":null,"authors":["Jesse Elliott"],"abstract":"This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notion of \"degree\" and a more general notion that we call \"logexponential degree.\" Specifically, we define the \\emph{degree} of a real function $f$ whose domain is not bounded above to be the infimum of all real numbers $t$ such that $f(x)$ is $O(x^t)$. The Riemann hypothesis, for example, is equivalent to the statement that the degree of the function $\\pi(x)- \\operatorname{li}(x)$ is $1/2$, where $\\pi(x)$ is the prime counting function and $\\operatorname{li}(x)$ is the logarithmic integral function; likewise, the abc conjecture is equivalent to the statement that a particular function has degree 1. Part 1 of the text is a survey of analytic number theory, Part 2 introduces the notion of logexponential degree and uses it to extend results in algebraic asymptotic analysis, and Part 3 applies the results of Part 2 to the various functions that figure most prominently in analytic number theory and Diophantine analysis. Central to the notion of logexponential degree are Hardy's \\emph{logarithmico-exponential functions}, which are real functions defined in a neighborhood of $\\infty$ that can be built from $\\operatorname{id}$, $\\exp$, and $\\log$ using the operations $+$, $\\cdot$, $/$, and $\\circ$. Such functions are natural benchmarks for the orders of growth of functions in analytic number theory. The main goal of Part 3 is to express the logexponential degree of various functions in analytic number theory in terms of as few \"logexponential primitives\" as possible.","url_abs":"https://arxiv.org/abs/2407.17820v4","url_pdf":"https://arxiv.org/pdf/2407.17820v4.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":"analytic-number-theory-and-algebraic","repo_url":"https://github.com/teorth/expdb","is_official":1,"mentioned_in_paper":1,"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}