On the average trigonometric sums for solutions of quadratic comparisons
https://doi.org/10.22405/2226-8383-2025-26-4-438-446
Abstract
In 1963, Hawley [1] (see also [2]), based on estimates of average trigonometric sums from solutions of quadratic comparisons, proved for the first time an asymptotic formula for the average number of divisors of a quadratic polynomial with a power-law decrease in the residual term. These results were later reinforced in [3] and [4]. In the present paper, we prove stronger results on this topic.
About the Authors
Viktor Alexeevich Bykovskii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Irina Yur’evna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
References
1. Hooley, C. 1963, “On the number of divisors of quadratic polynomials”, Acta Math., 110(1–2), 97–114.
2. Hooley, C. 1963, “On the number of divisors of quadratic polynomials”, Acta Math., 110(1–2), 97–114. (Russian translation: collection Mathematics, 1968, 12:5, pp. 3–18.)
3. Bykovsky, V. A. 1981, “Asymptotic properties of integer points (𝑎1, 𝑎2) satisfying the relation 𝑎1𝑎2 ≡ 𝑙(mod 𝑞)”, In Notes of Scientific Seminars of LOMI, L.: Nauka, 112, pp. 5–25.
4. Bykovsky, V. A. 1982, “On a summation formula in the spectral theory of automorphic functions and its applications in analytic number theory”, Dokl. AN USSR, 264(2), pp. 275–277.
5. Dirichlet, P. 1936, “Lectures on number theory”, M: Leningrad, 404 p.
6. Bateman, H., & Erdelyi, A. 1953, “Higher Transcendental Functions”, McGraw-Hill Book Company, Vol. 1, 2.
7. Conroy B., Iwaniec H. 2000, “The cubic moment of central values of automorphic L–functions”, Ann. of Math. (2) 151, no. 3, pp. 1175-1216.
8. Liu, S., & Hasri, R. 2015, “The average of the divisor function over values of quadratic polynomial”, Proc. Amer. Math. Soc., 143, pp. 4143–4160.
Review
For citations:
Bykovskii V.A., Rebrova I.Yu. On the average trigonometric sums for solutions of quadratic comparisons. Chebyshevskii Sbornik. 2025;26(4):438-446. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-4-438-446
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