Oscillations of a given size of some arithmetic error terms

Kaczorowski, Jerzy; Wiertelak, Kazimierz

doi:10.1090/S0002-9947-09-04803-X

Oscillations of a given size of some arithmetic error terms
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Trans. Amer. Math. Soc. 361 (2009), 5023-5039 Request permission

Abstract:

A general method of estimating the number of oscillations of a given size of arithmetic error terms is developed. Special attention is paid to the remainder terms in the prime number formula, in the Dirichlet prime number theorem for primes in arithmetic progressions and to the remainder term in the asymptotic formula for the number of square free divisors of an integer.

References

R. C. Baker, The square-free divisor problem, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 179, 269–277. MR 1295577, DOI 10.1093/qmath/45.3.269
R. C. Baker, The square-free divisor problem. II, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 186, 133–146. MR 1397933, DOI 10.1093/qmath/47.2.133
A. A. Gioia and A. M. Vaidya, The number of squarefree divisors of an integer, Duke Math. J. 33 (1966), 797–799. MR 202678, DOI 10.1215/S0012-7094-66-03396-5
J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. I, Acta Arith. 44 (1984), no. 4, 365–377. MR 777013, DOI 10.4064/aa-44-4-365-377
J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. I, Acta Arith. 44 (1984), no. 4, 365–377. MR 777013, DOI 10.4064/aa-44-4-365-377
J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. III, Acta Arith. 48 (1987), no. 4, 347–371. MR 927376, DOI 10.4064/aa-48-4-347-371
J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. IV, Acta Arith. 50 (1988), no. 1, 15–21. MR 945273, DOI 10.4064/aa-50-1-15-21
J. Kaczorowski and J. Pintz, Oscillatory properties of arithmetical functions. I, Acta Math. Hungar. 48 (1986), no. 1-2, 173–185. MR 858395, DOI 10.1007/BF01949062
J. Kaczorowski, The $k$-functions in multiplicative number theory. I. On complex explicit formulae, Acta Arith. 56 (1990), no. 3, 195–211. MR 1083000, DOI 10.4064/aa-56-3-195-211
J. Kaczorowski, The $k$-functions in multiplicative number theory. III. Uniform distribution of zeta zeros; discrepancy, Acta Arith. 57 (1991), no. 3, 199–210. MR 1105605, DOI 10.4064/aa-57-3-199-210
Jerzy Kaczorowski and Kazimierz Wiertelak, $\Omega$-estimates for a class of arithmetic error terms, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 3, 385–394. MR 2329690, DOI 10.1017/S0305004107000035
J.E. Littlewood, Sur la distribution des nombres premiers, C.R. Acad. Sci. Paris 158 (1914), 1869–1872.
F. Mertens, Über einige asymptotische Gesetze der Zahlentheorie, J. reine angew. Math. 77 (1874), 289–338.
H. L. Montgomery, The zeta function and prime numbers, Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979) Queen’s Papers in Pure and Appl. Math., vol. 54, Queen’s Univ., Kingston, Ont., 1980, pp. 1–31. MR 634679
Hugh L. Montgomery and Ulrike M. A. Vorhauer, Changes of sign of the error term in the prime number theorem, Funct. Approx. Comment. Math. 35 (2006), 235–247. MR 2271616, DOI 10.7169/facm/1229442626
Werner Georg Nowak and Michael Schmeier, Conditional asymptotic formulae for a class of arithmetic functions, Proc. Amer. Math. Soc. 103 (1988), no. 3, 713–717. MR 947644, DOI 10.1090/S0002-9939-1988-0947644-0
János Pintz, Cramér vs. Cramér. On Cramér’s probabilistic model for primes. part 2, Funct. Approx. Comment. Math. 37 (2007), no. part 2, 361–376. MR 2363833, DOI 10.7169/facm/1229619660
Bahman Saffari, Sur le nombre des diviseurs “$r$-libres” d’un entier, et sur les points à coordonnées entières dans certaines régions du plan, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A601–A603 (French). MR 230692
D. Suryanarayana and V. Siva Rama Prasad, The number of $k$-free divisors of an integer, Acta Arith. 17 (1970/71), 345–354. MR 327697, DOI 10.4064/aa-17-4-345-354
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
Kazimierz Wiertelak, On some connections between zeta-zeros and square-free divisors of an integer, Funct. Approx. Comment. Math. 31 (2003), 133–145. MR 2059542, DOI 10.7169/facm/1538186644
Jie Wu, On the primitive circle problem, Monatsh. Math. 135 (2002), no. 1, 69–81. MR 1894296, DOI 10.1007/s006050200006