Additive arithmetic functions meet the inclusion-exclusion principle
| Autor: | Bordell��s, Olivier, T��th, L��szl�� |
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| Rok vydání: | 2021 |
| Předmět: | |
| DOI: | 10.48550/arxiv.2104.07443 |
| Popis: | We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x} f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le x} f([n_1,\ldots,n_k])$ involving the gcd and lcm of the integers $n_1,\ldots,n_k$, where $f$ belongs to certain classes of additive arithmetic functions. In particular, we consider the generalized omega function $��_{\ell}(n)= \sum_{p^��\mid\mid n} ��^{\ell}$ investigated by Duncan (1962) and Hassani (2018), and the functions $A(n)=\sum_{p^��\mid\mid n} ��p$, $A^*(n)= \sum_{p \mid n} p$, $B(n)=A(n)-A^*(n)$ studied by Alladi and Erd��s (1977). As a key auxiliary result we use an inclusion-exclusion-type identity. 22 pages |
| Databáze: | OpenAIRE |
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