Oscillations of the remainder term related to the Euler totient function
Let \(\varphi(n)\) be Euler's function and let \[ E(x)=\sum_{n\leq x}\varphi(n)-\frac{3}{\pi^2}x^2. \] Let \(\mu(n)\) be Möbius' function and let \[ f(x)=-\sum_{n=1}^\infty\frac{\mu(n)}{n}\left\{\frac{x}{n}\right\}, \] \[ g(x)=\sum_{n=1}^\infty\mu(n)\left\{\frac{x}{n}\right\}^2, \] where \(\{\theta\}\) is the fractional part of a real number \(\theta\). The authors prove for \(x\geq 1\): \[ E(x)=xf(x)+\frac12 g(x)+\frac12. \] They write \[ E(x)=E^{\text{AR}}(x)+E^{\text{AN}}(x),E^{\text{AR}}(x)=xf(x),E^{\text{AN}}(x)=\frac12 g(x)+\frac12 \] and show \[ \begin{aligned} f(x)&=\Omega_{\pm}(\sqrt{\log\log x}),\\ E^{\text{AR}}(x)&=\Omega_{\pm}(x\sqrt{\log\log x}),\\ E^{\text{AN}}(x)&=\Omega_{\pm}(x^\frac12\log\log\log x),\end{aligned} \] as \(x\to\infty\). The following statements are equivalent: (1) The Riemann Hypothesis is true; (2) There exists a positive constant \(A\) such that for \(x\geq e^e\) we have \[ E^{\text{AN}}(x)\ll x^\frac12\exp\left(A\frac{\log x}{\log\log x}\right); \] (3) For every \(\varepsilon>0\) and \(x\geq 1\) we have \[ E^{\text{AN}}(x)\ll_\varepsilon x^{\frac12 + \varepsilon}. \]
scientific article; zbMATH DE number 5802579
scientific article