Abstract:
In the article we have for any fixed $c$ estimate of $\kappa(c)$ such that $N>N_0(\varepsilon)$ can be approached by the sum of powers of two primes $p_1^c+p_2^c$ by a distance not exceeding $H=N^{\kappa(c)+\varepsilon}$, where $\varepsilon$ is an arbitrary positive number.
These results were obtained using the density technique developed by Yu.V. Linnik in the 1940s. The density technique is based on applying explicit formulas expressing sums over prime numbers with sums over nontrivial zeros of the Riemann zeta function and using density theorems that estimate the number of nontrivial zeros of the zeta function lying in the critical strip such that their real part is greater than some $\sigma$, $1> \sigma \geq 1/2$.
The results obtained in this paper are based on the application of modern density theorems obtained by A. Ivich. In addition, the proof used the theorem of Baker, Harman, and Pintz: one can approach a given real number $N>N_0(\varepsilon)$ by a prime number by a distance no more than $H = N^{21/40 + \varepsilon}$. Also, the following result obtained by M. Huxley: $|\zeta(1/2+it)|\ll t^{32/205+\varepsilon}$.
Keywords:primes, diophantine inequalities, density theorem.
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