On the constant factor in several related asymptotic estimates
Author:
Andreas Weingartner
Journal:
Math. Comp. 88 (2019), 1883-1902
MSC (2010):
Primary 11N25, 11Y60
DOI:
https://doi.org/10.1090/mcom/3402
Published electronically:
December 26, 2018
MathSciNet review:
3925489
Full-text PDF
View in AMS MathViewer
Abstract | References | Similar Articles | Additional Information
Abstract: We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.
- [1] Christian Axler, New estimates for some functions defined over primes, Integers 18 (2018), Paper No. A52, 21. MR 3813836
- [2] P. Dusart, Estimates of Some Functions Over Primes without R.H., arXiv:1002.0442 (2018).
- [3] Rudolf Lidl and Harald Niederreiter, Finite fields, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. MR 1429394
- [4] Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, J. Number Theory 37 (1991), no. 1, 1–36 (French, with English summary). MR 1089787, https://doi.org/10.1016/S0022-314X(05)80022-8
- [5] 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
- [6] Karl K. Norton, Upper bounds for sums of powers of divisor functions, J. Number Theory 40 (1992), no. 1, 60–85. MR 1145854, https://doi.org/10.1016/0022-314X(92)90028-N
- [7] Carl Pomerance, Lola Thompson, and Andreas Weingartner, On integers 𝑛 for which 𝑋ⁿ-1 has a divisor of every degree, Acta Arith. 175 (2016), no. 3, 225–243. MR 3557122
- [8] G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. (9) 63 (1984), no. 2, 187–213 (French, with English summary). MR 774171
- [9] Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR 1876657
- [10] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689
- [11] Eric Saias, Entiers à diviseurs denses. I, J. Number Theory 62 (1997), no. 1, 163–191 (French, with English summary). MR 1430008, https://doi.org/10.1006/jnth.1997.2057
- [12] J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥), Math. Comp. 29 (1975), 243–269. MR 457373, https://doi.org/10.1090/S0025-5718-1975-0457373-7
- [13] Waclaw Sierpiński, Sur une propriété des nombres naturels, Ann. Mat. Pura Appl. (4) 39 (1955), 69–74 (French). MR 75219, https://doi.org/10.1007/BF02410762
- [14] A. K. Srinivasan, Practical numbers, Current Sci. 17 (1948), 179–180. MR 0027799
- [15] B. M. Stewart, Sums of distinct divisors, Amer. J. Math. 76 (1954), 779–785. MR 64800, https://doi.org/10.2307/2372651
- [16] 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
- [17] Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, 2nd ed., Cours Spécialisés [Specialized Courses], vol. 1, Société Mathématique de France, Paris, 1995 (French). MR 1366197
- [18] Lola Thompson, Polynomials with divisors of every degree, J. Number Theory 132 (2012), no. 5, 1038–1053. MR 2890525, https://doi.org/10.1016/j.jnt.2011.10.005
- [19] Andreas Weingartner, Practical numbers and the distribution of divisors, Q. J. Math. 66 (2015), no. 2, 743–758. MR 3356847, https://doi.org/10.1093/qmath/hav006
- [20] Andreas Weingartner, On the degrees of polynomial divisors over finite fields, Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 3, 469–487. MR 3569157, https://doi.org/10.1017/S030500411600044X
- [21] Andreas Weingartner, A sieve problem and its application, Mathematika 63 (2017), no. 1, 213–229. MR 3610011, https://doi.org/10.1112/S002557931600022X
Retrieve articles in Mathematics of Computation with MSC (2010): 11N25, 11Y60
Retrieve articles in all journals with MSC (2010): 11N25, 11Y60
Additional Information
Andreas Weingartner
Affiliation:
Department of Mathematics, 351 West University Boulevard, Southern Utah University, Cedar City, Utah 84720
Email:
weingartner@suu.edu
DOI:
https://doi.org/10.1090/mcom/3402
Received by editor(s):
May 17, 2017
Received by editor(s) in revised form:
July 17, 2018
Published electronically:
December 26, 2018
Article copyright:
© Copyright 2018
American Mathematical Society