Abstract
A Beurling generalized number system is constructed having integer counting function N
B
(x) = κ
x +O(x
θ) with κ>0 and 1/2 <θ <1, whose prime counting function satisfies the oscillation estimate π
B
(x) =li(x) + Ω(xexp(-c
)), and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ). The construction uses elements of classical analytic number theory and probability.
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References
Apostol, T.M.: Mathematical Analysis. Addison–Wesley, Reading, MA, 1957
Bateman, P.T., Diamond, H.G.: Asymptotic distribution of Beurling's generalized prime numbers. In: Studies in Number Theory, W.J. LeVeque (ed.), Math. Assoc. Am., Washington, D. C., (distributed by Prentice-Hall, Englewood Cliffs, N.J.) 1969, pp. 152–210
Beurling, A.: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I. Acta Math. 68, 255–291 (1937)
Burgess, D.A.: The distribution of quadratic residues and non-residues. Mathematika 4, 106–112 (1957)
Diamond, H.G.: Asymptotic distribution of Beurling's generalized integers. Illinois J. Math. 14, 12–28 (1970)
Friedlander, J., Iwaniec, H.: Estimates for character sums. Proc. Am. Math. Soc. 119, 365–372 (1993)
Granville, A., Soundararajan, K.: Notes on Burgess's theorem. To appear
Hall, R.S.: Theorems about Beurling's generalized primes and the associated zeta function. Ph.D. Thesis, Univ. of Illinois, Urbana, 1967
Landau, E.: Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes. Math. Annalen 56, 645–670 (1903); Collected Works Vol. 1, Thales Verlag, Essen, pp. 327–352
Landau, E.: Neue Beiträge zur analytischen Zahlentheorie. Rend. Circ. Mat. Palermo 27, 46–58 (1909); Collected Works Vol. 4, Thales Verlag, Essen, pp. 41–53
Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen. Second Edition, Chelsea, New York, 1953
Loève, M.M.: Probability theory. Fourth Edition, Springer–Verlag, New York, 1977
Malliavin, P.: Sur le reste de la loi asymptotique de répartition des nombres premiers généralisés de Beurling. Acta Math. 106, 281–298 (1961)
Montgomery, H.L.: Topics in Multiplicative Number Theory. Lecture Notes in Math., Vol. 227, Springer, Berlin Heidelberg, 1971
Petrov, V.V.: Limit theorems of probability: sequences of independent random variables. Oxford University Press, Oxford, 1995
Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function. Second Edition D.R. Heath-Brown (ed.), The Clarendon Press, Oxford University Press, New York, 1986
Turán, P.: On the remainder-term of the prime number formula, II. Acta Math. Acad. Sci. Hungar. 1, 155–166 (1950)
de la Vallée Poussin, Ch.J.: Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée. Mémoires couronnés et autres mémoires publiés par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique 59(1), 74 pp (1899–1900)
Weber, H.: Über einen in der Zahlentheorie angewandten Satz der Integralrechnung. Nachr. Akad. Wiss. Göttingen, 1896, pp. 275–281
Zhang, W.-B.: Asymptotic distribution of Beurling's generalized prime numbers and integers. Ph.D. Thesis, Univ. of Illinois, Urbana, 1986
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The author was supported in part by NSF grants DMS-0070720 and DMS-0244660.
The author was supported in part by NSF grant DMS-0244660.
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Diamond, H., Montgomery, H. & Vorhauer, U. Beurling primes with large oscillation. Math. Ann. 334, 1–36 (2006). https://doi.org/10.1007/s00208-005-0638-2
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DOI: https://doi.org/10.1007/s00208-005-0638-2