Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences

[2509.24456] Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2509.24456 (math) [Submitted on 29 Sep 2025 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences Authors: Giovanni Coppola View PDF HTML (experimental) Abstract: In this self-contained short note, we introduce the new definition of Good Ramanujan Expansion, say G.R.E., for a fixed arithmetic function $F$, building upon a good decay of its coefficients $G$; this, gains $\log-$powers w.r.t. the trivial bound for $G$ and precisely $\log^{1+\eta}$, where the present parameter $\eta>0$ is real. This property alone has important consequences for all the $F$ having a G.R.E. : mainly, 1) the Eratosthenes Transform $F'$ of our $F$ is infinitesimal (see in Theorem 1); 2) when $\eta>1$ (an enhanced decay) we have uniqueness of $G$ (actually, these are the classic Wintner-Carmichael coefficients, see Th.2); 3) we get a bound for $F$ (in Th.3); 4) an important new class of arithmetic functions $F$ can't have a G.R.E. (see Th.4). These are a generalization of Correlations; which in this way, if are, say, a kind of "far from constants", may not have a G.R.E., whence, a fortiori, can't have the R.E.E.F. This is the Ramanujan Exact Explicit Formula, that we introduced with Prof. Ram Murty. On the Hardy-Ramanujan Journal, I proved that: any "fair" Correlation may be "well-approximated" by a (BH) Correlation. For a (BH) Correlation, we prove here: it has a G.R.E. IF AND ONLY IF it has the (R.E.E.F.). On the other hand, next Counterexample 1 in section 3 is a (BH) Correlation without the (R.E.E.F.); also, it provides here a (BH) Correlation behaving like in Theorem 4. Comments: One page more, one reference more (Cohen's paper), one Theorem more (the fifth) Subjects: Number Theory (math.NT) MSC classes: 11N37 Cite as: arXiv:2509.24456 [math.NT] (or arXiv:2509.24456v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2509.24456 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Giovanni Coppola [ view email ] [v1] Mon, 29 Sep 2025 08:37:59 UTC (13 KB) [v2] Sun, 31 May 2026 09:18:33 UTC (15 KB) [v3] Fri, 28 Aug 2026 09:08:06 UTC (16 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-09 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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