Multiplicative functions on arithmetic progressions. VI: More middle moduli
[Part V, cf. J. Lond. Math. Soc., II. Ser. 41, No. 3, 408-424 (1990; Zbl 0671.10045).] The principal result of this paper is the following theorem on the distribution of general multiplicative functions on arithmetic progressions: Let \(g\) be a complex-valued multiplicative function satisfying \(| g|\leq 1\), and let \(0<\beta<1\), \(0<\delta<1/2\), \(2\leq\log N\leq Q\leq N\) be given. Then \[ \sum_{\substack{n\leq x \\ n\equiv r\pmod D}} g(n)= \frac1{\varphi(D)} \sum _{\substack{n\leq x \\ (n,d)=1}} g(n)+O \left( \frac{x} {\varphi(D)} \biggl( \frac{\log Q}{\log x} \biggr)^{1/8-\delta} \right) \] holds for \(N^\beta \leq x\leq N\), \((r,D)=1\), for all moduli \(D\leq Q\) save possibly the multiples of some \(D_ 0>1\), where the \(O\)- constant depends at most on \(\beta\) and \(\delta\). This estimate represents a significant improvement, both in terms of the size of the error term and the range of uniformity in the modulus \(D\), of earlier results by the author [Mathematika 34, 199--206 (1987; Zbl 0637.10032)] and the reviewer [Proc. Am. Math. Soc. 108, 307--318 (1990; Zbl 0692.10038)], and is probably close to best-possible under the given general hypotheses. The uniformity in the modulus \(D\) is the same as in Gallagher's prime number theorem [\textit{P. Gallagher}, Invent. Math. 11, 329--339 (1970; Zbl 0219.10048)]. In fact, the author shows that Gallagher's theorem in the form \[ \pi(x,D,r)= \frac{\pi(x)}{\varphi(D)} \left( 1+O \Biggl( \biggl( \frac{\log Q}{\log x}\biggr)^{1/8-\delta} \Biggl)\right), \] where \(x\), \(Q\), and \(D\) are as above, can be deduced from his result, thus providing a new proof of Gallagher's theorem (with a somewhat weaker error term) which avoids the use of density results for \(L\)-functions.
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