Abstract
We obtain global explicit numerical bounds, with best possible constants, for the differences \(\frac{1}{n}\sum _{k\leqslant n}\omega (k)-\log \log n\) and \(\frac{1}{n}\sum _{k\leqslant n}\Omega (k)-\log \log n\), where \(\omega (k)\) and \(\Omega (k)\) refer to the number of distinct prime divisors, and the total number of prime divisors of k, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We mention that the Maple command to compute \(\Omega (n)\) is bigomega(n) and accordingly, a Maple code to compute \(\omega (n)\) is given by
-
with(numtheory):
-
rad:= n -> convert(numtheory:-factorset(n), ‘*‘):
-
smallomega:=n->bigomega(rad(n));
-
References
Apostol, T.M.: Mathematical analysis, Second edition, Addison-Wesley Publishing Company, (1974)
Diaconis, P.: Asymptotic expansions for the mean and variance of the number of prime factors of a number \(n\), Technical Report No. 96, Department of Statistics, Stanford University, December 14, (1976)
Duncan, R.L.: A class of additive arithmetical functions. Amer. Math. Monthly 69, 34–36 (1962)
Finch, S.R.: Mathematical constants, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2003)
Hardy, G., Ramanujan, S.: The normal number of prime factors of a number \(n\). Quart. J. Math. 48, 76–92 (1917)
Hassani, M.: Factorization of factorials and a result of Hardy and Ramanujan. Math. Inequal. Appl. 15, 403–407 (2012)
Hassani, M.: On the difference of the number-theoretic omega functions. J. Combin. Number Theory 8, 165–178 (2016)
Hassani, M.: Remarks on the number of prime divisors of integers. Math. Inequal. Appl. 16, 843–849 (2013)
Hassani, M.: Asymptotic expansions for the average of the generalized omega function. Integers 18, 23 (2018)
Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bull. Amer. Math. Soc. 50, 527–628 (2013)
Littlewood, J.E.: Sur la distribution des nombres premiers. Comptes Rendus 158, 1869–1872 (1914)
Saffari, B.: Sur quelques applications de la “méthode de l’hyperbole’’ de Dirichlet à la théorie des nombres premiers. Enseignement Math. 14, 205–224 (1970)
Schoenfeld, L.: Sharper bounds for the Chebyshev functions \(\theta (x)\) and \(\psi (x)\). II, Math. Comput., 30, 337–360 (1976)
Tenenbaum, G.: Introduction to analytic and probabilistic number theory, Third edition, American Mathematical Society, (2015)
Trudgian, T.: Updating the error term in the prime number theorem. Ramanujan J. 39, 225–234 (2016)
von Koch, H.: Sur la distribution des nombres premiers. Acta Math. 24, 159–182 (1901)
Acknowledgements
The author is greatly indebted to Prof. Horst Alzer for suggesting the problem of finding global numerical bounds for the number-theoretic omega functions and for many stimulating conversations. Also, he is greatly indebted to the referee for a thorough reading of the manuscript and helpful comments.
Author information
Authors and Affiliations
Contributions
Mehdi Hassani did all parts of the paper.
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hassani, M. Global Numerical Bounds for the Number-Theoretic Omega Functions. Mediterr. J. Math. 20, 319 (2023). https://doi.org/10.1007/s00009-023-02527-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-023-02527-7