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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.10636 |
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| _version_ | 1866908768992755712 |
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| author | Alamoudi, Yazan |
| author_facet | Alamoudi, Yazan |
| contents | In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,ω}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } μ(n) {ω(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+ε}})$ with $Y_0,\mathscr{p},ε>0$. Moreover, I give preliminary upper bounds for the general range $1.9\leq y\leq x^{\frac{1}{k}}$. In addition, I formalize the notions of subradical and radical dominance and discuss their relevance to the analytic approach of the study of arithmetic functions. Lastly, I give a fascinating formula related to the derivatives of the gamma function and the Hankel contour, which should be relevant for those employing the Selberg--Delange method to obtain higher-order terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10636 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On subradically sifted sums related to Alladi's higher order duality between prime factors Alamoudi, Yazan Number Theory In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,ω}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } μ(n) {ω(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+ε}})$ with $Y_0,\mathscr{p},ε>0$. Moreover, I give preliminary upper bounds for the general range $1.9\leq y\leq x^{\frac{1}{k}}$. In addition, I formalize the notions of subradical and radical dominance and discuss their relevance to the analytic approach of the study of arithmetic functions. Lastly, I give a fascinating formula related to the derivatives of the gamma function and the Hankel contour, which should be relevant for those employing the Selberg--Delange method to obtain higher-order terms. |
| title | On subradically sifted sums related to Alladi's higher order duality between prime factors |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.10636 |