## 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

### MSC:

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

### Citations:

Zbl 0052.279; Zbl 0044.039
Full Text:

### References:

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