References
- M. R. Albrecht, R. Player, S. Scott, On the concrete hardness of learning with errors, J. Math. Cryptol. 9 (2019), 169–203.
- T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
- A. Bailleul, L. Devin, D. Keliher, W. Li, Exceptional biases in counting primes over function fields, J. London Math. Soc. 109 (2024), Article e12876.
- P. Billingsley, Probability and Measure, Wiley, 1995.
- P. Borwein, S. K. K. Choi, M. Coons, Completely multiplicative functions taking values in {-1, 1}, Trans. Amer. Math. Soc. 362 (2010), 6279–6291.
- 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.
- J. Chinis, On the Liouville Function in Short Intervals, Int. Math. Res. Not. IMRN 2022 (15) (2022), 11203–11219.
- H. Daboussi, A. Sárközy, On pseudorandom properties of multiplicative functions, Acta Math. Hungar. 98 (2003), 273–300.
- H. Daboussi, A. Sárközy, On the correlation of the truncated Liouville function, Acta Arith. 108 (2003), 61–76.
- 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.
- R. Durrett, Probability: Theory and Examples, Cambridge University Press, 2010.
- P. Erdős, M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738–742.
- G. H. Hardy, S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Math. 48 (1917), 76–92.
- 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.
- J. Katz, Y. Lindell, Introduction to Modern Cryptography (3rd ed.), CRC Press, 2020.
- J. Kubilius, Probabilistic Methods in the Theory of Numbers, AMS, 1964.
- M. J. Mossinghoff, T. S. Trudgian, Oscillations in weighted arithmetic sums, Int. J. Number Theory 17 (2021), 1697–1716.
- J. Rivat, A. Sárközy, C. L. Stewart, Congruence properties of the Ω-function on sumsets, Illinois J. Math. 43 (1999), 1–18.
- A. Sankaranarayanan, On the estimation of nonlinear twists of the Liouville function, Illinois J. Math. 56 (2) (2012), 551–569.
- G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, American Mathematical Society, Providence, RI, 2015.
- V. V. Williams, On some fine-grained questions in algorithms and complexity, in Proceedings of the International Congress of Mathematicians (2019), 3447–3487.
Language: English
Page range: 45 - 61
Submitted on: Oct 3, 2025
Accepted on: Dec 20, 2025
Published on: Jul 11, 2026
Published by: Ovidius University of Constanta
In partnership with: Paradigm Publishing Services
Keywords:
Related subjects:
© 2026 Ahmed Gaber, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.