Diamond's version of the Dirichlet hyperbola method
The paper is devoted to Beurling numbers. Consider a sequence of generalized prime numbers \(1<\beta_1\leq\beta_2\leq\dots \), \(\lim_{n\to+\infty}\beta_n=+\infty\) and the corresponding generalized integer numbers \(\beta^\nu:=\beta_1^{\nu_1}\beta_2^{\nu_2}\dots.\) Let \[ P(x):=\sum_{\beta_n\leq x}1\quad \text{and}\quad N(x):=\sum_{\beta^\nu\leq x}1. \] In the theory of generalized numbers, relations between equalities \[ P(x)=\text{li}(x)+x\eta(x)\quad \text{and}\quad N(x)=Dx+x\varepsilon(x), \] where \(D\) is some positive constant, are usually investigated. In [Ill. J. Math. 14, 12-28 (1970; Zbl 0186.36403)] \textit{H. Diamond} stated without a proof that if \(\eta(x)=O(\log^{-a} x)\), \(a>0\), then \(\varepsilon(x)=O(\log^{3-a} x)\). The author gives a proof of this result. Moreover, using some modification of the Diamond method, he proves that if \(\beta_n=p_n+O(n^a)\) with \(0\leq a<1\), where \(p_n\) is a prime number, then \(\varepsilon(x)\ll\exp\left(-c\sqrt{\log x\log\log x}\right)\) with \(c<\sqrt{2(1-a)}\).
scientific article; zbMATH DE number 1558748