On effective mean-values of arithmetic functions
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917152837074944 |
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| author | Tenenbaum, Gérald |
| author_facet | Tenenbaum, Gérald |
| contents | Let $r,\,f$ be multiplicative functions with $r\geqslant 0$, $f$ is complex valued, $|f|\leqslant r$, and $r$ satisfies some standard growth hypotheses. Let $x$ be large, and assume that, for some real number $τ$, the quantities $r(p)-\Re\{f(p)/p^{iτ}\}$ are small in various appropriate average senses over the set of prime numbers not exceeding $x$. We derive from recent effective mean-value estimates an effective comparison theorem between the mean-values of $f$ and of $r$ on the set of integers $\leqslant x$. We also provide effective estimates for certain weighted moments of additive functions and for sifted mean-values of non-negative multiplicative functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On effective mean-values of arithmetic functions Tenenbaum, Gérald Number Theory primary 11N37, secondary 11N25, 11N60, 11N64 Let $r,\,f$ be multiplicative functions with $r\geqslant 0$, $f$ is complex valued, $|f|\leqslant r$, and $r$ satisfies some standard growth hypotheses. Let $x$ be large, and assume that, for some real number $τ$, the quantities $r(p)-\Re\{f(p)/p^{iτ}\}$ are small in various appropriate average senses over the set of prime numbers not exceeding $x$. We derive from recent effective mean-value estimates an effective comparison theorem between the mean-values of $f$ and of $r$ on the set of integers $\leqslant x$. We also provide effective estimates for certain weighted moments of additive functions and for sifted mean-values of non-negative multiplicative functions. |
| title | On effective mean-values of arithmetic functions |
| topic | Number Theory primary 11N37, secondary 11N25, 11N60, 11N64 |
| url | https://arxiv.org/abs/2507.10483 |