On the mean values of the Chebyshev function and their applications
https://doi.org/10.22405/2226-8383-2021-22-5-198-222
Abstract
Assuming the validity of the extended Riemann hypothesis for the average values of Chebyshev functions over all characters modulo 𝑞, the following estimate holds
$$𝑡(𝑥; 𝑞) =Σ︁𝜒mod𝑞max𝑦≤𝑥|𝜓(𝑦, 𝜒)| ≪ 𝑥 + 𝑥1/2𝑞L2, L = ln 𝑥𝑞$$.
When solving a number of problems in prime number theory, it is sufficient that 𝑡(𝑥; 𝑞) admits an estimate close to this one. The best known estimates for 𝑡(𝑥; 𝑞) previously belonged to
G. Montgomery, R. Vaughn, and Z. Kh. Rakhmonov. In this paper we obtain a new estimate of the form
$$𝑡(𝑥; 𝑞) = Σ︁ 𝜒mod𝑞 max 𝑦≤𝑥 |𝜓(𝑦, 𝜒)| ≪ 𝑥L^28 + 𝑥^(4/5) 𝑞^(1/2)L^31 + 𝑥^(1/2)𝑞L^32$$,
using which for a linear exponential sum with primes we prove a stronger estimate $$𝑆(𝛼, 𝑥) ≪ 𝑥𝑞^(−1/2)L^33 + 𝑥^(4/5)L^32 + 𝑥^(1/2)𝑞^(1/2)L^33$$, when $$|𝑎-a/q|<1/q^2, (a,q)=1$$.
We also study the distribution of Hardy-Littlewood numbers of
the form 𝑝+𝑛2 in short arithmetic progressions in the case when the difference of the progression is a power of the prime number.
About the Authors
Zarullo Khusenovich Rakhmonov
National Academy of Sciences of Tajikistan, A. Dzhuraev Institute of Mathematics
Uzbekistan
Uzbekistan
doctor of physical and mathematical sciences, professor,
academician
Opokkhon Okilkhonovich Nozirov
A. Dzhuraev Institute of Mathematics
Uzbekistan
Uzbekistan
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Review
For citations:
Rakhmonov Z.Kh., Nozirov O.O. On the mean values of the Chebyshev function and their applications. Chebyshevskii Sbornik. 2021;22(5):198-222. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-5-198-222
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