Fluctuations in the mean of Euler’s phi function. (English) Zbl 0656.10042

Define the remainder term R(x) by \(\sum_{n\leq x}\phi (n)=3/\pi^ 2\cdot x^ 2+R(x)\), where \(\phi\) (n) denotes Euler’s totient function. A. Z. Walfisz [Tr. Tbilis. Mat. Inst. Razmadze 19, 1-31 (1953; Zbl 0052.279)] gave the upper bound \[ R(x)\ll x(\log x)^{2/3}(\log \log x)^{4/3}, \] while S. D. Chowla and S. S. Pillai [J. Lond. Math. Soc. 5, 95-101 (1930)] showed that \(R(x)=\Omega (x \log \log \log x).\) P. Erdős and H. N. Shapiro [Can. J. Math. 3, 375-385 (1951; Zbl 0044.039)] demonstrated that R(x) changes sign infinitely often by proving \(R(x)=\Omega_{\pm}(x \log \log \log \log x).\)
The present paper provides the estimate \[ (*)\quad R(x)=\Omega_{\pm}(x (\log \log x)^{1/2}), \] thereby improving on both of the results of Erdős and Shapiro and of Chowla and Pillai. In order to prove (*) the author uses complex integration to obtain the following refinement of a result of Chowla and Pillai (loc. cit.): \[ (**)\quad R_ 0(x)=R(x)/x+O(\exp (-c \log^{1/2}x)), \] where \(R_ 0(x)\) is defined by \(\sum_{n\leq x}\phi (n)/n=6/\pi^ 2\cdot x+R_ 0(x)\). \(R_ 0(x)\) may be represented by sums involving the Möbius function and the function s(x) which has period 1 and satisfies \(s(0)=0\), \(s(x)=-x\) for \(0<x<1\). This can be used to evaluate \(\sum_{n\leq N}R_ 0(qn+\alpha),\) where q is a positive integer not exceeding exp(c \(log^{1/2}N)\) and \(\alpha\) is a non-integral real number, \(0<\alpha <q\). A suitable choice of q and \(\alpha\) yields (*).
Furthermore the author conjectures that \(R(x)\ll x \log \log x\) and \(R(x)=\Omega_{\pm}(x \log \log x)\).
Reviewer: J.Herzog


11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
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