Arithmetic and Asymptotic Properties of Restricted Totient Sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918137646022656 |
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| author | En-naoui, Es-said |
| author_facet | En-naoui, Es-said |
| contents | This article extends our previous study on the summatory behavior of Euler's totient function $φ(n)$. We investigate two complementary restricted sums, $Υ(x,p)=\sum_{\substack{k\le x\\\gcd(k,p)=1}}φ(k)$ and $Δ(x,p)=\sum_{\substack{k\le x\\p\mid k}}φ(k)$, which satisfy the decomposition $Ψ(x)=\sum_{k\le x}φ(k)=Υ(x,p)+Δ(x,p)$. We establish recurrence formulas, congruence relations, and generating function identities for $Δ(x,p)$. In particular, we prove that $Δ(x,p)\equiv 0\pmod{p-1}$ for every prime $p$, and we derive the asymptotic expansion $Δ(x,p)=\dfrac{3}{π^{2}(p+1)}\,x^{2}+O(x\log x)$. Furthermore, we study average orders, connections with $ω(n)$, and relations with divisor structures. These results refine the analytic understanding of totients in arithmetic progressions and complement the classical asymptotic theory of $Ψ(x)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic and Asymptotic Properties of Restricted Totient Sums En-naoui, Es-said General Mathematics 11N37, 11A25 This article extends our previous study on the summatory behavior of Euler's totient function $φ(n)$. We investigate two complementary restricted sums, $Υ(x,p)=\sum_{\substack{k\le x\\\gcd(k,p)=1}}φ(k)$ and $Δ(x,p)=\sum_{\substack{k\le x\\p\mid k}}φ(k)$, which satisfy the decomposition $Ψ(x)=\sum_{k\le x}φ(k)=Υ(x,p)+Δ(x,p)$. We establish recurrence formulas, congruence relations, and generating function identities for $Δ(x,p)$. In particular, we prove that $Δ(x,p)\equiv 0\pmod{p-1}$ for every prime $p$, and we derive the asymptotic expansion $Δ(x,p)=\dfrac{3}{π^{2}(p+1)}\,x^{2}+O(x\log x)$. Furthermore, we study average orders, connections with $ω(n)$, and relations with divisor structures. These results refine the analytic understanding of totients in arithmetic progressions and complement the classical asymptotic theory of $Ψ(x)$. |
| title | Arithmetic and Asymptotic Properties of Restricted Totient Sums |
| topic | General Mathematics 11N37, 11A25 |
| url | https://arxiv.org/abs/2509.07004 |