×

A proof of a conjecture of Bateman and Diamond on Beurling generalized primes. (English) Zbl 1321.11107

Author’s abstract: Beurling’s generalized prime system is a sequence \(\{p_i\}\) of real numbers \(p_i\) satisfying \(1<p_1\leq p_2\leq \cdots\), \(p_i \to \infty\). The multiplicative semigroup \(\{n_1=1<n_2\leq n_3\leq \cdots\}\) generated by \(\{p_i\}\) is called a system of Beurling’s generalized integers. If \(\pi(x)\) and \(N(x)\) denote the counting functions of generalized primes \(p_i\) and of generalized integers \(n_i\), respectively, Beurling’s problem is to find conditions on \(N(x)\) which imply “the prime number theorem”, i.e., \(\pi(x)\sim x/\log x\) as \(x\to \infty\). Assuming that \(N(x)=Ax+xE(x)\), Beurling’s condition is \(E(x)=O(\log^{-\gamma}x)\) with \(\gamma >3/2\); but \(E(x)=O(\log^{-3/2}x)\) does not suffice. Bateman and Diamond conjectured the condition \(\int (E(x)\log x)^2 x^{-1} dx<\infty\). A proof of this conjecture following J.-P. Kahane’s method [J. Théor. Nombres Bordx. 9, No. 2, 251–266 (1997; Zbl 0905.11042)] is given in this paper. The proof is based on Poission’s summation formula and the Wiener-Ikehara tauberian theorem and applies classical ideas.

MSC:

11N80 Generalized primes and integers
11M45 Tauberian theorems
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Citations:

Zbl 0905.11042
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bateman, P.T., Diamond, H.G.: Asymptotic distribution of Beurling’s generalized primes, Studies in Number Theory, Vol. 6, 152-210, Math. Assoc. Am., Prentice-Hall, Englewood Cliffs (1969) · Zbl 0216.31403
[2] Bateman, P.T., Diamond, H.G.: Analytic Number Theory: An Introductory Course. World Scientific, Singapore (2004) · Zbl 1074.11001
[3] Beurling, A.: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I. Acta Math. 68, 255-291 (1937) · JFM 63.0138.01 · doi:10.1007/BF02546666
[4] Diamond, H.G.: A set of generalized numbers showing Beurling’s theorem to be sharp. Illinois J. Math. 14, 29-34 (1970) · Zbl 0186.36501
[5] Diamond, H.G.: The prime number theorem for Beurling’s generalized numbers. J. Number Theory 1, 200-207 (1969) · Zbl 0167.32001 · doi:10.1016/0022-314X(69)90038-9
[6] Kahane, J.P.: Une formule de Fourier sur les nombres premiers. Gazette des Mathématiciens 67, 3-9 (1996). Janvier · Zbl 0879.11001
[7] Kahane, J.P.: Une formule de Fourier pour nombres premiers généralisés de Beurling, Publ. Math. Orsay 96.1 (1996), “Harmonic analysis from the Pichorides viewpoint” (Myriam Déchamps, ed.) · Zbl 0874.11068
[8] Kahane, J.P.: A Fourier formula for prime numbers. Can. Math. Soc. Conf. Proc. 21, 89-102 (1997) · Zbl 0905.11041
[9] Kahane, J.P.: Sur les nombres premiers généralisés de Beurling, Preuve d’une conjecture de Bateman et Diamond. J. de Théorie des Nombres de Bordeaux 9, 251-266 (1997) · Zbl 0905.11042 · doi:10.5802/jtnb.201
[10] Kahane, J.P.: Le rôle des algebres A de Wiener, \[A^\infty\] A∞ de Beurling et \[H^1\] H1 de Sobolev dans la théorie des nombres premiers généralisés. Ann. Inst. Fourier (Grenoble) 48(3), 611-648 (1998) · Zbl 0905.11043 · doi:10.5802/aif.1632
[11] Schlage, J.-C., Vindas, J.: The prime number theorem for Beurling’s generalized numbers. New cases, Acta Arith. 133(3), 293-324 (2012) · Zbl 1300.11099
[12] Zhang, W.-B.: Wiener-Ikehara theorems and the Beurling generalized primes. Monatsh. Math. doi:10.1007/s00605-013-0597-8 · Zbl 1384.11090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.