On the mean values of the Chebyshev function and their applications

Assuming the validity of the extended Riemann hypothesis for the average values of Chebyshev functions over all characters modulo $q$, the following estimate holds
$$ t(x;q)=\sum_{\chi\mod q}\max_{y\leq x}|\psi(y,\chi)|\ll x+x^{1/2}q\mathscr{L}^2,\quad \mathscr{L}=\ln xq. $$
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
$$ t(x;q)=\sum_{\chi\mod q}\max_{y\leq x}|\psi(y,\chi)|\ll x\mathscr{L}^{28}+x^{\frac{4}{5}}q^{\frac12}\mathscr{L}^{31}+x^\frac{1}{2}q\mathscr{L}^{32}, $$
using which for a linear exponential sum with primes we prove a stronger estimate
$$ S(\alpha,x)\ll xq^{-\frac12}\mathscr{L}^{33}+x^{\frac{4}{5}}\mathscr{L}^{32}+x^\frac{1}{2}q^\frac12\mathscr{L}^{33}, $$
when $\left|\alpha-\frac aq\right|<\frac1{q^2}$, $(a,q)=1$. 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.