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2013 | 11 | 3 | 477-486
Tytuł artykułu

On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

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Języki publikacji
EN
Abstrakty
EN
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.
Twórcy
  • Institute of Mathematics, Department of Integrative Biology, BOKU Wien, 1180, Vienna, Austria, kleitner@boku.ac.at
autor
  • Institute of Mathematics, Department of Integrative Biology, BOKU Wien, 1180, Vienna, Austria, nowak@boku.ac.at
Bibliografia
  • [1] Balasubramanian R., Ramachandra K., Subbarao M.V., On the error function in the asymptotic formula for the counting function of k-full numbers, Acta Arith., 1988, 50(2), 107–118
  • [2] Huxley M.N., Area, Lattice Points, and Exponential Sums, London Math. Soc. Monogr. (N.S.), 13, Oxford University Press, New York, 1996
  • [3] Huxley M.N., Exponential sums and lattice points. III, Proc. London Math. Soc., 2003, 87(3), 591–609 http://dx.doi.org/10.1112/S0024611503014485[Crossref]
  • [4] Huxley M.N., Exponential sums and the Riemann zeta-function. V, Proc. London Math. Soc., 2005, 90(1), 1–41 http://dx.doi.org/10.1112/S0024611504014959[Crossref]
  • [5] Ivic A., The Riemann Zeta-Function, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1985
  • [6] Krätzel E., Lattice Points, Math. Appl. (East European Ser.), 33, Kluwer, Dordrecht, 1988
  • [7] Krätzel E., Nowak W.G., Tóth L., On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 2012, 10(2), 761–774 http://dx.doi.org/10.2478/s11533-011-0144-6[Crossref][WoS]
  • [8] Krätzel E., Nowak W.G., Tóth L., On a class of arithmetic functions connected with a certain asymmetric divisor problem, In: 20th Czech and Slovak International Conference on Number Theory, Stará Lesná, September 5–9, 2011 (abstracts), Slovak Academy of Sciences, Bratislava, 14–15
  • [9] Kühleitner M., An Omega theorem on Pythagorean triples, Abh. Math. Sem. Univ. Hamburg, 1993, 63, 105–113 http://dx.doi.org/10.1007/BF02941336[Crossref]
  • [10] Kühleitner M., Nowak W.G., An Omega theorem for a class of arithmetic functions, Math. Nachr., 1994, 165, 79–98 http://dx.doi.org/10.1002/mana.19941650107[Crossref]
  • [11] Kühleitner M., Nowak W.G., The average number of solutions of the Diophantine equation U 2 + V 2 = W 3 and related arithmetic functions, Acta Math. Hungar., 2004, 104(3), 225–240 http://dx.doi.org/10.1023/B:AMHU.0000036284.91580.3e[Crossref]
  • [12] Montgomery H.L., Vaughan R.C., Hilbert’s inequality, J. London Math. Soc., 1974, 8, 73–82 http://dx.doi.org/10.1112/jlms/s2-8.1.73[Crossref]
  • [13] Prachar K., Primzahlverteilung, Springer, Berlin-Göttingen-Heidelberg, 1957
  • [14] Ramachandra K., A large value theorem for ζ(s), Hardy-Ramanujan J., 1995, 18, 1–9
  • [15] Ramachandra K., Sankaranarayanan A., On an asymptotic formula of Srinivasa Ramanujan, Acta Arith., 2003, 109(4), 349–357 http://dx.doi.org/10.4064/aa109-4-5[Crossref]
  • [16] Schinzel A., On an analytic problem considered by Sierpinski and Ramanujan, In: New Trends in Probability and Statistics, 2, Palanga, 1991, VSP, Utrecht, 1992, 165–171
  • [17] Sloane N., On-Line Encyclopedia of Integer Sequences, #A055155, http://oeis.org/A055155
  • [18] Sloane N., On-Line Encyclopedia of Integer Sequences, #A078430, http://oeis.org/A078430
  • [19] Sloane N., On-Line Encyclopedia of Integer Sequences, #A124316, http://oeis.org/A124316
  • [20] Soundararajan K., Omega results for the divisor and circle problems, Int. Math. Res. Notices, 2003, 36, 1987–1998 http://dx.doi.org/10.1155/S1073792803130309[Crossref]
  • [21] Szegö G., Beiträge zur Theorie der Laguerreschen Polynome. II: Zahlentheoretische Anwendungen, Math. Z., 1926, 25, 388–404 http://dx.doi.org/10.1007/BF01283847[Crossref]
  • [22] Titchmarsh E.C., The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford University Press, Oxford, 1986
  • [23] Tóth L., Menon’s identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 2011, 69(1), 97–110
  • [24] Tóth L., Weighted gcd-sum functions, J. Integer Seq., 2011, 14(7), #11.7.7
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0143-2
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