Hildebrand, Adolf Quantitative mean value theorems for nonnegative multiplicative functions. II. (English) Zbl 0573.10034 Acta Arith. 48, 209-260 (1987). [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 Cited in 5 ReviewsCited in 4 Documents MSC: 11N60 Distribution functions associated with additive and positive multiplicative functions 11N35 Sieves 11M06 \(\zeta (s)\) and \(L(s, \chi)\) Keywords:mean value theorems; lower estimate; multiplicative functions; asymptotic estimates; Dickman function; Erdős-Ruzsa conjecture; lower bound sieve estimates Citations:Zbl 0307.10041; Zbl 0338.10032; Zbl 0573.10034 PDFBibTeX XMLCite \textit{A. Hildebrand}, Acta Arith. 48, 209--260 (1987; Zbl 0573.10034) Full Text: DOI EuDML