An extension of the Siegel-Walfisz theorem

Weingartner, Andreas

doi:10.1090/proc/15607

An extension of the Siegel-Walfisz theorem
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by Andreas Weingartner

Proc. Amer. Math. Soc. 149 (2021), 4699-4708

DOI: https://doi.org/10.1090/proc/15607

Published electronically: August 4, 2021

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Abstract:

We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on the size of the prime factors. Besides prime powers, this family includes smooth numbers, almost primes and practical numbers.

References

J. B. Friedlander, Selberg’s formula and Siegel’s zero, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 15–23. MR 637340
Adam J. Harper, On a paper of K. Soundararajan on smooth numbers in arithmetic progressions, J. Number Theory 132 (2012), no. 1, 182–199. MR 2843307, DOI 10.1016/j.jnt.2011.07.005
Emmanuel Knafo, On a generalization of the Selberg formula, J. Number Theory 125 (2007), no. 2, 319–343. MR 2332592, DOI 10.1016/j.jnt.2006.10.013
G. Kolesnik, On the order of Dirichlet $L$-functions, Pacific J. Math. 82 (1979), no. 2, 479–484. MR 551704
Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, J. Number Theory 37 (1991), no. 1, 1–36 (French, with English summary). MR 1089787, DOI 10.1016/S0022-314X(05)80022-8
Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
C. Pomerance and A. Weingartner, On primes and practical numbers, Ramanujan J. (2021), https://doi.org/10.1007/s11139-020-00354-y.
Eric Saias, Entiers à diviseurs denses. I, J. Number Theory 62 (1997), no. 1, 163–191 (French, with English summary). MR 1430008, DOI 10.1006/jnth.1997.2057
A. K. Srinivasan, Practical numbers, Current Sci. 17 (1948), 179–180. MR 27799
Gérald Tenenbaum, Sur un problème de crible et ses applications, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 1–30 (French). MR 860809
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995. Translated from the second French edition (1995) by C. B. Thomas. MR 1342300
Andreas Weingartner, Practical numbers and the distribution of divisors, Q. J. Math. 66 (2015), no. 2, 743–758. MR 3356847, DOI 10.1093/qmath/hav006
Andreas Weingartner, A sieve problem and its application, Mathematika 63 (2017), no. 1, 213–229. MR 3610011, DOI 10.1112/S002557931600022X
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp. 88 (2019), no. 318, 1883–1902. MR 3925489, DOI 10.1090/mcom/3402
Andreas Weingartner, The constant factor in the asymptotic for practical numbers, Int. J. Number Theory 16 (2020), no. 3, 629–638. MR 4079398, DOI 10.1142/S1793042120500311