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

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 $\ddot{I}{†}_{\ell }$ by \[ φ_\ell(n):=n\prod_{\substack{p\ \mathrm{prime}\\ v_p(n)\ge \ell}}\left(1-\frac{1}{p}\right), \] and its summatory function by \[ Φ_\ell(x):=\sum_{n\le x}φ_\ell(n). \] An analytic study of the generalized Euler $\ell$-totient function $\ddot{I}{†}_{\ell }(n)$ 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 $\hat{I}{¦}_{\ell }(x)$.

16 pages

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