×

Beurling’s generalized prime numbers and Sobolev’s algebra \(H^ 1\). (Le rôle de l’algèbre \(H^ 1\) de Sobolev dans la théorie des nombres premiers généralisés de Beurling.) (French) Zbl 0886.11058

Let \((p_i)_{i\in\mathbb{N}}\) be a sequence of generalized primes, i.e. \(p_i\) are real numbers and \(1<p_1\leq p_2\leq\cdots, p_i\rightarrow\infty\), and \((n_i)_{i\in\mathbb{N}_0}\) be the associated free semi-group of Beurling integers. Define for \(x>0\) the counting functions \[ N(x) := \sum_{n_i \leq x} 1; \qquad P(x) := \sum_{p_i\leq x} 1. \] A. Beurling [Acta Math. 68, 255-291 (1937; Zbl 0017.29604)] proved that if \(N(x) = A x + O\bigl(x \varepsilon(x)\bigr)\) for some constants \(A >0\) and \(\varepsilon(x)=O\bigl(\log^{-\gamma}x\bigr)\) with \(\gamma > \frac32\) then the generalized prime number theorem (PNT) \[ P(x) \sim \frac{x}{\log x} \quad (x\to\infty) \] holds. H. G. Diamond [Ill. J. Math. 14, 29-34 (1970; Zbl 0186.36501)] showed that \(\gamma = 3/2\) does not suffice. It is an old conjecture of Bateman and Diamond that the condition \(\int_1^\infty \bigl(\varepsilon(x)\log x)^2\frac{\text{ d}x}{x} <\infty\) implies the PNT. In the note under review an analytical proof of this conjecture is sketched. Using the associated \(\zeta\)-function defined by \[ \zeta(s)=\frac{A}{s-1} +A-1 +s\int_1^\infty x^{-s}\varepsilon(x)\text{ d}x, \] for \(\text{Re } s>1\) it is easy to see that \((s-1)\zeta(s)\) could be continued to the half-plane \(\text{Re } s\geq1\). The main step is to show that \(it\zeta(1+it)\) multiplied with a suitable weight function belongs to the Sobolev algebra \(H^1\).

MSC:

11N80 Generalized primes and integers
PDFBibTeX XMLCite
Full Text: DOI