A generalization of Halász's theorem to Beurling's generalized integers and its application
Halász's theorem [\textit{G. Halász}, Acta Math. Acad. Sci. Hung. 19, 365-403 (1968; Zbl 0165.058)] is generalized to Beurling's generalized integers in this paper. Let \({\mathcal P}=\{p_ i\}^{\infty}_{i=1}\) be a set of generalized primes in Beurling's sense and \({\mathcal N}=\{n_ i\}^{\infty}_{i=0}\) the set of generalized integers associated with \({\mathcal P}\). \(N(x)=\sum _{n_ i\leq x}1\) denotes the distribution function of the generalized integers in \({\mathcal N}\). The author proves the following three theorems: (1) Let \(f(n_ i)\) be a completely multiplicative function on \({\mathcal N}\) such that \(| f(n_ i)| \leq 1\) holds for all \(n_ i\in {\mathcal N}\). Suppose that for some constant \(A>0\), \(\int ^{\infty}_{1}x^{- 2} | N(x)-Ax | dx<\infty\) and either \(\int ^{x}_{1}t^{- 1}\{N(t)-At\}\log t dt \ll x\) or \(\int ^{\infty}_{1}x^{-3} | N(x)-Ax | ^ 2 \log x dx<\infty\) holds. Furthermore, suppose that \(\hat F(s):=\sum ^{\infty}_{i=0}f(n_ i)/n^ s_ i=c/(s- 1)+o(1/(\sigma -1))\) holds as \(\sigma =Re s\to 1+\) uniformly for -K\(\leq t\leq K\) for each fixed \(K>0\). Then we have \[ F(x):=\sum _{n_ i\leq x}f(n_ i)=cx+o(x)\quad. \] (2) Suppose that the conditions on \(f(n_ i)\) and N(x) are satisfied. Then \(F(x)=o(x)\) holds if and only if \(\sum ^{\infty}_{i=1}p_ k^{-1} Re(1-f(p_ k)p_ k^{-it})=\infty\) holds for all real t. (3) Suppose that the conditions on N(x) are satisfied. Then \(M(x):=\sum _{n_ i\leq x}\mu (n_ i)=o(x)\), where \(\mu (n_ i)\) is the anlogue of the classical Möbius function. The last result combined with a theorem of \textit{A. Beurling} [Acta Math. 68, 255-291 (1937; Zbl 0017.29604)] and an example of \textit{H. G. Diamond} [Ill. J. Math. 14, 12-28, 29-34 (1970; Zbl 0186.364, Zbl 0186.365)] shows that the prime number theorem and the assertion that \(M(x)=o(x)\) are not completely equivalent.
scientific article; zbMATH DE number 3989467