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Quantitative mean value theorems for nonnegative multiplicative functions. II. (English) Zbl 0573.10034

[Part I, cf. J. Lond. Math. Soc., II. Ser. 30, 394–406 (1985; Zbl 0573.10034).]
A lower estimate for sums of nonnegative multiplicative functions is given. It is shown, in particular, that uniformly for \(x\geq 2\) and all multiplicative functions \(f\) satisfying \(0\leq f\leq 1\) the estimate \[ \frac1x\sum_{n\leq x}f(n)\geq \prod_{p\leq x}\left(1-\frac1p\right)\left(1+\sum_{m\geq 1}\frac{f(p^ m)}{p^ m}\right)\, \sigma \left(\exp \left(\sum_{p\leq x}\frac{(1-f(p))}{p}\right)\right)\times \]
\[ \times\left(1+O\left((\log x)^{-\alpha}\right)\right)+O\left(\exp (-(\log x)^{\beta})\right) \] holds. Here \(\alpha\) and \(\beta\) are positive constants and \(\sigma (u)=u \rho(u)\), where \(\rho\) is Dickman’s function. This is applied to prove a conjecture of Erdős and Ruzsa on a lower bound sieve estimate.
As a further application, it is shown that for a real, non-principal character \(\chi\) modulo \(D\) the bound \[ \exp \left(\sum_{{p\leq D^ 2}\atop {\chi (p)=-1}}\frac1p\right)\gg \frac{\log (1/L(1,\chi))}{\log \log (1/L(1,\chi))} \] holds, provided \(L(1,\chi)\leq 1/\log^ 2 D\). This improves on a result of J. Pintz [Acta Arith. 31, 273–289 (1976; Zbl 0307.10041)].
Reviewer: A. Hildebrand

MSC:

11N60 Distribution functions associated with additive and positive multiplicative functions
11N35 Sieves
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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