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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0143-2,
author = {Manfred K\"uhleitner and Werner Nowak},
title = {On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {477-486},
zbl = {1294.11166},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0143-2}
}
Manfred Kühleitner; Werner Nowak. On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions. Open Mathematics, Tome 11 (2013) pp. 477-486. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0143-2/
[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 | Zbl 0652.10033
[2] Huxley M.N., Area, Lattice Points, and Exponential Sums, London Math. Soc. Monogr. (N.S.), 13, Oxford University Press, New York, 1996 | Zbl 0861.11002
[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] | Zbl 1065.11079
[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] | Zbl 1083.11052
[5] Ivic A., The Riemann Zeta-Function, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1985 | Zbl 0556.10026
[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] | Zbl 1285.11120
[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] | Zbl 0795.11045
[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] | Zbl 0835.11036
[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] | Zbl 1060.11058
[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] | Zbl 1036.11045
[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 | Zbl 0767.11045
[17] Sloane N., On-Line Encyclopedia of Integer Sequences, #A055155, http://oeis.org/A055155 | Zbl 1274.11001
[18] Sloane N., On-Line Encyclopedia of Integer Sequences, #A078430, http://oeis.org/A078430 | Zbl 1274.11001
[19] Sloane N., On-Line Encyclopedia of Integer Sequences, #A124316, http://oeis.org/A124316 | Zbl 1274.11001
[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] | Zbl 1130.11329
[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] | Zbl 52.0175.03
[22] Titchmarsh E.C., The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford University Press, Oxford, 1986 | Zbl 0601.10026
[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 | Zbl 1235.11011
[24] Tóth L., Weighted gcd-sum functions, J. Integer Seq., 2011, 14(7), #11.7.7 | Zbl 1259.11014