On some generalized integers
Let \((q_i)\) be a sequence of generalized primes, i.e. \( q_i\) are real numbers and \( 1 = q_0 <q_1\leq q_2 \leq \cdots\), \(q_i \to \infty\), and \( (n_i)_{i \in \mathbb N_0} \) be the associated set of Beurling integers. Define for \(x >0\) the counting functions \[ B(x) :=\sum_{n_i \leq x} 1; \quad P(x) := \sum_{q_i\leq x}1 . \] \textit{A. Beurling} [Acta Math. 68, 255-291 (1937; Zbl 0017.29604)] started the study of such number systems. Essentially, two problems were investigated: 1) to deduce an asymptotic formula for \(P(x)\) and, in particular, an analogon of the prime number theorem \(P(x) \sim x/\log x\) from asymptotics of \(B(x)\), and 2) in the other direction to prove an asymptotic formula for \(B(x)\) under certain assumptions on the behavior of the generalized primes. In the paper under review the author considers a problem of the second kind: for given \(r>0\) define generalized primes \(q_i\) by \(q_i:= p_{m+i}-r\), \(i=1,2,\ldots\), where \(m=\pi(r)+1\) and \(p_n\) is the ordinary \(n\)-th rational prime number. Then there exists an explicitly given constant \(b(r)>0\) and a remainder \(R(x)\) of order roughly \(O(x\exp(-\log^{1/2}x))\) such that \[ B(x)=b(r)x+R(x). \] The case \(r=1\) is connected with the well studied sum \(\sum_{\varphi(n)\leq x}1\), where \(\varphi\) denotes Euler's function. In the proofs the author uses the classical methods of contour integration. As he remarked, his remainder term (in the case \(r=1\)) is not as sharp as a result which M. Balazard obtained quite recently.
scientific article; zbMATH DE number 1059803
scientific article