On the mean values of the Chebyshev functions and their applications

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 $$t(x;q)=\sum _{\chi q}\underset{y\le x}{\max }\mathrm{\mid }\psi (y,\chi )\mathrm{\mid }\ll x{\mathcal{L}}^{28}+x^{\frac{4}{5}}{q}^{\frac{1}{2}}{\mathcal{L}}^{31}+x^{\frac{1}{2}}q{\mathcal{L}}^{32},$$ using which for a linear exponential sum with primes we prove a stronger estimate $$S(\alpha ,x)\ll x{q}^{-\frac{1}{2}}{\mathcal{L}}^{33}+x^{\frac{4}{5}}{\mathcal{L}}^{32}+x^{\frac{1}{2}}{q}^{\frac{1}{2}}{\mathcal{L}}^{33},$$ when $\mid \alpha -\frac{a}{q}\mid <\frac{1}{q^2}$, $(a,q)=1$. 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.