Author:
János Pintz ELKH Alfréd Rényi Mathematical Institute, Reáltanoda u. 13-15, H-1053 Budapest, Hungary

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In the 1980’s the author proved lower bounds for the mean value of the modulus of the error term of the prime number theorem and other important number theoretic functions whose oscillation is in connection with the zeros of the Riemann zeta function. In the present work a general theorem is shown in a simple way which gives a lower bound for the mentioned mean value as a function of a hypothetical pole of the Mellin transform of the function. The conditions are amply satisfied for the Riemann zeta function. In such a way the results recover the earlier ones (even in a slightly sharper form). The obtained estimates are often optimal apart from a constant factor, at least under reasonable conditions as the Riemann Hypothesis. This is the case, in particular, for the error term of the prime number theorem.

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    Brent, R. P., Platt, D. J., and Trudgian, T. S. The mean square of the error term in the prime number theorem. arXiv:2008.06140.

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    Cramèr, H. Some theorems concerning prime numbers. Arkivf. Math. Astr. Fys. 15, 5 (1920), 1-33.

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    Knapowski, S. On the mean values of certain functions in prime number theory. Acta Math. Acad. Sci. Hungar 10 (1959), 375-390.

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    Kátai, I. On oscillations of number-theoretic functions. Acta Arith. 13 (1967/68), 107-122.

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    Landau, E. Über einen Satz von Tschebyschef. Math. Ann. 61 (1905), 527-550.

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    Phragmèn, E. Sur le logarithme intègral et la fonction f(x) de Riemann. Öfversigt Kongl. Vet.-Akad. Förh. Stockholm 48 (1891), 599-616.

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  • [7]

    Pintz, J. On the remainder term of the prime number formula. I. On a problem of Littlewood. Acta Arith. 36, 4 (1980), 341-365.

  • [8]

    Pintz, J. Oscillatory properties of M x = n x μ n. I. Acta Arith. 42, 1 (1982/83), 49-55.

  • [9]

    Pintz, J. On the distribution of square-free numbers. J. London Math. Soc. (2) 28, 3 (1983), 401-405.

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    Pintz, J. On the mean value of the remainder term of the prime number formula. In Elementary and analytic theory of numbers (Warsaw, 1982). Banach Center Publ., 17, PWN, Warsaw, 1985, p. 411-417.

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  • [11]

    Révész, Sz. Gy. Effective oscillation theorems for a general class of real-valued remainder terms. Acta Arith. 49, 5 (1988), 481-505.

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  • [12]

    Schmidt, E. Über die Anzahl der Primzahlen unter gegebener Grenze. Mat. Ann. 57 (1903), 195-204.

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    Turán, P. On the remainder-term of the prime-number formula. I. Acta Math. Acad. Sci. Hungar. 1 (1950), 48-63.

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    Turán, P. Eine neue Methode in der Analysis und deren Anwendungen. Akadémiai Kiadó, Budapest, 1953. In German.

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    Turán, P. On a new method of analysis and its applications. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. xvi+584 pp. With the assistance of G. Halász and J. Pintz. With a foreword by Vera T. Sós. Pure and Applied Mathematics (New York).

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Editor in Chief: László TÓTH (University of Pécs)

Honorary Editors in Chief:

  • † István GYŐRI (University of Pannonia, Veszprém)
  • János PINTZ (Rényi Institute of Mathematics)
  • Ferenc SCHIPP (Eötvös University Budapest and University of Pécs)
  • Sándor SZABÓ (University of Pécs)
     

Deputy Editors in Chief:

  • Erhard AICHINGER (JKU Linz)
  • Ferenc HARTUNG (University of Pannonia, Veszprém)
  • Ferenc WEISZ (Eötvös University, Budapest)

Editorial Board

  • György DÓSA (University of Pannonia, Veszprém)
  • István BERKES (Rényi Institute of Mathematics)
  • Károly BEZDEK (University of Calgary)
  • Balázs KIRÁLY – Managing Editor (University of Pécs)
  • Vedran KRCADINAC (University of Zagreb) 
  • Željka MILIN ŠIPUŠ (University of Zagreb)
  • Margit PAP (University of Pécs)
  • Mihály PITUK (University of Pannonia, Veszprém)
  • Jörg THUSWALDNER (Montanuniversität Leoben)
  • Zsolt TUZA (University of Pannonia, Veszprém)

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  • Gabriella BÖHM (Akadémiai Kiadó, Budapest)
  • György GÁT (University of Debrecen)

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Mathematica Pannonica
Language English
Size A4
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