On the approximation of real numbers by the sums of square of primes

Abstract

In the article it is proved that a given real number $N>N_0(\varepsilon)$ can be approached by the sum of squares of three primes by a distance not exceeding $H = N^{217/768 + \varepsilon}$ and can be approached by the sum of four squares of primes by a distance no greater than $H = N^{1519/9216 + \varepsilon}$, 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 the author is used: one can approach a given real number $N>N_0(\varepsilon)$ by the sum of squares of two prime numbers by a distance no greater than $H = N^{31/64 + \varepsilon}$.