Parity results concerning the generalized divisor function involving small prime factors of integers
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arXiv
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| Format: | Preprint |
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2024
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| author | Alladi, Krishnaswami Goswami, Ankush |
| author_facet | Alladi, Krishnaswami Goswami, Ankush |
| contents | Let $ν_y(n)$ denote the number of distinct prime factors of $n$ that are
$<y$. For $k$ a positive integer, and for $k+2\leq y\leq x$, let $S_{-k}(x,y)$ denote the sum \begin{eqnarray*} S_{-k}(x,y):=\sum_{n\leq x}(-k)^{ν_y(n)}. \end{eqnarray*} In this paper, we describe our recent results on the asymptotic behavior of $S_{-k}(x,y)$ for $k+2\leq y\leq x$, and $x$ sufficiently large. There is a crucial difference in the asymptotic behavior of $S_{-k}(x,y)$ when $k+1$ is a prime and $k+1$ is composite, and this makes the problem particularly interesting. The results are derived utilizing a combination of the Buchstab-de Bruijn recurrence, the Perron contour integral method, and certain difference-differential equations. We present a summary of our results against the background of earlier work of the first author on sums of the Möbius function over integers with restricted prime factors and on a multiplicative generalization of the sieve. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parity results concerning the generalized divisor function involving small prime factors of integers Alladi, Krishnaswami Goswami, Ankush Number Theory 11M06, 11M41, 11N37, 11N25, 11N35 Let $ν_y(n)$ denote the number of distinct prime factors of $n$ that are $<y$. For $k$ a positive integer, and for $k+2\leq y\leq x$, let $S_{-k}(x,y)$ denote the sum \begin{eqnarray*} S_{-k}(x,y):=\sum_{n\leq x}(-k)^{ν_y(n)}. \end{eqnarray*} In this paper, we describe our recent results on the asymptotic behavior of $S_{-k}(x,y)$ for $k+2\leq y\leq x$, and $x$ sufficiently large. There is a crucial difference in the asymptotic behavior of $S_{-k}(x,y)$ when $k+1$ is a prime and $k+1$ is composite, and this makes the problem particularly interesting. The results are derived utilizing a combination of the Buchstab-de Bruijn recurrence, the Perron contour integral method, and certain difference-differential equations. We present a summary of our results against the background of earlier work of the first author on sums of the Möbius function over integers with restricted prime factors and on a multiplicative generalization of the sieve. |
| title | Parity results concerning the generalized divisor function involving small prime factors of integers |
| topic | Number Theory 11M06, 11M41, 11N37, 11N25, 11N35 |
| url | https://arxiv.org/abs/2412.03088 |