Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences
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2025
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| author | Coppola, Giovanni |
| author_facet | Coppola, Giovanni |
| contents | 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 $\log^{1+η}$, where the present parameter $η>0$ 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 $η>1$ (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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_24456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences Coppola, Giovanni Number Theory 11N37 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 $\log^{1+η}$, where the present parameter $η>0$ 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 $η>1$ (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. |
| title | Good Ramanujan Expansions: A suitably enhanced decay of coefficients has important consequences |
| topic | Number Theory 11N37 |
| url | https://arxiv.org/abs/2509.24456 |