On the mean values of the Chebyshev functions and their applications
Z. Rakhmonov, O. Nozirov
TL;DR
The paper advances the theory of mean values of Chebyshev functions by proving a sharper bound for t(x;q) that combines x, q, and logarithmic factors, and then propagates this improvement to stronger estimates for linear exponential sums with primes S(\alpha,x) near rational frequencies. It develops a robust toolkit of Main Lemmas to bound complex sums t_k(q;M,N) and related Dirichlet-L-function expressions, enabling precise control over prime-sum distributions. Consequently, it derives a new bound for S(\frac{a}{q},x) and demonstrates corollaries for arbitrary real frequencies α via rational approximations. Finally, it applies these methods to Hardy-Littlewood numbers, proving an asymptotic formula in very short progressions modulo prime powers with explicit error terms and identifying the range of nontrivial results. These results tighten classic bounds by Montgomery, Vaughan, and Rakhmonov and have implications for prime distribution and additive problems in short intervals and progressions.
Abstract
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 $$ t(x;q)=\sum_{χ\bmod q}\max_{y\leq x}|ψ(y,χ)|\ll x{\mathscr{L}}^{28}+x^\frac45q^\frac12{\mathscr{L}}^{31}+x^\frac12q{\mathscr{L}}^{32}, $$ using which for a linear exponential sum with primes we prove a stronger estimate $$ S(α,x)\ll xq^{-\frac12}{\mathscr{L}}^{33}+x^\frac45{\mathscr{L}}^{32}+x^\frac12q^\frac12{\mathscr{L}}^{33}, $$ when $\left|α-\frac aq\right|<\frac1{q^2}$, $(a,q)=1$. We also study the distribution of Hardy-Littlewood numbers of the form $ p + n ^ 2 $ in short arithmetic progressions in the case when the difference of the progression is a power of the prime number. Bibliography: 30 references.