Short quadratic exponential sums with primes in major arcs
https://doi.org/10.22405/2226-8383-2025-26-3-220-234
Abstract
Using the second moment of Dirichlet 𝐿-functions on the critical line over the major arcs M(L𝑏), with 𝜏 = 𝑦3𝑥−1L−𝑏1 , and excluding a small neighborhood of the centers of these arcs, i.e., those 𝛼 satisfying |𝛼 − 𝑎
𝑞 | > (8𝜋𝑦2)−1, for 𝑦 ⩾ 𝑥^1− 1/(9−4√2)L𝑐2 , where 𝑐2 = 2𝐴+24+(√2−1)𝑏1/2√2−1, we obtain the estimate
Moreover, in a small neighborhood of the center of the major arcs, defined by |𝛼−𝑎𝑞 | ⩽ (8𝜋𝑦2)−1,for 𝑦 ⩾ 𝑥^5/8 L1,5𝐴+0,25𝑏+18, an asymptotic formula with a remainder term is obtained for 𝑆2(𝛼; 𝑥, 𝑦), where 𝐴, 𝑏1, and 𝑏 are arbitrary fixed positive constants, and L = ln 𝑥.
About the Authors
Zarullo Khusenovich Rakhmonov
Tajik National University
Tajikistan
Tajikistan
doctor of physical and mathematical sciences, Academician of the National Academy of Sciences of Tajikistan
Rakhbaroy Latifovna Khotamova
A. Dzhuraev Institute of Mathematics, National Academy of Sciences of Tajikistan
Tajikistan
Tajikistan
research associate
Mashrafi Sulaimon Sharifzoda
A. Dzhuraev Institute of Mathematics, National Academy of Sciences of Tajikistan
Tajikistan
Tajikistan
research associate
References
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Review
For citations:
Rakhmonov Z.Kh., Khotamova R.L., Sharifzoda M.S. Short quadratic exponential sums with primes in major arcs. Chebyshevskii Sbornik. 2025;26(3):220-234. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-3-220-234
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