ℓ-generalized prime factors counting functions

Ahmed Gaber

doi:10.2478/auom-2026-0019

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ℓ-generalized prime factors counting functions

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 34 (2026): Issue 2 (July 2026)

By: Ahmed Gaber

Open Access

|Jul 2026

Abstract

To study the number of extra prime factors beyond the first ℓ − 1 occurrences of a prime p in the factorization of an integer n, this paper introduces the ℓ-generalized prime factors counting arithmetic function Ω_A,ℓ. Key results include the Dirichlet series, an exact summatory formula, and the asymptotic mean. In addition, a distributional limit result is derived using Kubilius model and Kolmogorov three sreies theorem.

References

M. R. Albrecht, R. Player, S. Scott, On the concrete hardness of learning with errors, J. Math. Cryptol. 9 (2019), 169–203.
Search in Google Scholar Back to article
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
Search in Google Scholar Back to article
A. Bailleul, L. Devin, D. Keliher, W. Li, Exceptional biases in counting primes over function fields, J. London Math. Soc. 109 (2024), Article e12876.
Search in Google Scholar Back to article
P. Billingsley, Probability and Measure, Wiley, 1995.
Search in Google Scholar Back to article
P. Borwein, S. K. K. Choi, M. Coons, Completely multiplicative functions taking values in {-1, 1}, Trans. Amer. Math. Soc. 362 (2010), 6279–6291.
Search in Google Scholar Back to article
J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat, A. Sárközy, On finite pseudorandom binary sequences III: The Liouville function, I, Acta Arith. 87 (1999), 367–390.
Search in Google Scholar Back to article
J. Chinis, On the Liouville Function in Short Intervals, Int. Math. Res. Not. IMRN 2022 (15) (2022), 11203–11219.
Search in Google Scholar Back to article
H. Daboussi, A. Sárközy, On pseudorandom properties of multiplicative functions, Acta Math. Hungar. 98 (2003), 273–300.
Search in Google Scholar Back to article
H. Daboussi, A. Sárközy, On the correlation of the truncated Liouville function, Acta Arith. 108 (2003), 61–76.
Search in Google Scholar Back to article
J. De Koninck, L. Germán, I. Kátai, On the convolution of the Liouville function under the existence of Siegel zeros, Lith.Math.J. 55 (3) (2015), 331–342.
Search in Google Scholar Back to article
R. Durrett, Probability: Theory and Examples, Cambridge University Press, 2010.
Search in Google Scholar Back to article
P. Erdős, M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738–742.
Search in Google Scholar Back to article
G. H. Hardy, S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Math. 48 (1917), 76–92.
Search in Google Scholar Back to article
P. Humphries, S. M. S., A. W. Tian, Biases in prime factorizations and Liouville functions for arithmetic progressions, J. Théor. nombres Bordeaux 31 (1) (2019), 1–25.
Search in Google Scholar Back to article
J. Katz, Y. Lindell, Introduction to Modern Cryptography (3rd ed.), CRC Press, 2020.
Search in Google Scholar Back to article
J. Kubilius, Probabilistic Methods in the Theory of Numbers, AMS, 1964.
Search in Google Scholar Back to article
M. J. Mossinghoff, T. S. Trudgian, Oscillations in weighted arithmetic sums, Int. J. Number Theory 17 (2021), 1697–1716.
Search in Google Scholar Back to article
J. Rivat, A. Sárközy, C. L. Stewart, Congruence properties of the Ω-function on sumsets, Illinois J. Math. 43 (1999), 1–18.
Search in Google Scholar Back to article
A. Sankaranarayanan, On the estimation of nonlinear twists of the Liouville function, Illinois J. Math. 56 (2) (2012), 551–569.
Search in Google Scholar Back to article
G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, American Mathematical Society, Providence, RI, 2015.
Search in Google Scholar Back to article
V. V. Williams, On some fine-grained questions in algorithms and complexity, in Proceedings of the International Congress of Mathematicians (2019), 3447–3487.
Search in Google Scholar Back to article