Euler's $\ell$-totients and Riemann hypothesis

Ahmed Gaber

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arXiv · 2607.26114

Euler's $\ell$-totients and Riemann hypothesis

Ahmed Gaber

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Abstract

This paper develops a new analytic framework for investigating the Riemann hypothesis. For each fixed integer $\ell \ge 1$, define Euler's $\ell$-totient function $\varphi_\ell$ by \[ \varphi_\ell(n):=n\prod_{\substack{p\ \mathrm{prime}\\ v_p(n)\ge \ell}}\left(1-\frac{1}{p}\right), \] and its summatory function by \[ \Phi_\ell(x):=\sum_{n\le x}\varphi_\ell(n). \] An analytic study of the generalized Euler $\ell$-totient function $\varphi_\ell(n)$ is carried out, including its Euler product representation, meromorphic continuation, and pole structure. For each $\ell$, necessary and sufficient criteria for the Riemann hypothesis are established in terms of the asymptotic behavior of $\Phi_\ell(x)$.

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Ahmed Gaber. 2026-07-28. Euler's $\ell$-totients and Riemann hypothesis. https://arxiv.org/abs/2607.26114

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