About one additive problem Hua Loo Keng's

I. Allakov ; A. Sh. Safarov

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 32-45

Résumé

Let $X$ be enough big real number and $ k\geq2$ be a natural number, $M$ be a set of natural numbers $n$ not exceeding $X$, which cannot be written as a sum of prime and fixed degree a prime, $E_k (X)=\mathrm{card} M.$ In present paper is proved theorem. Theorem. For it is enough greater $X-$equitable estimation $ E_k (X)\ll X^{\gamma},$ where $$ \gamma\left\{ \begin{array}{lll} 1-(17612,983k^2 (\ln k+6,5452))^{-1}, \text{при} 2\leq k\leq 205,\\[1mm] 1-(68k^3 (2\ln k+\ln\ln k+2,8))^{-1}, \text{при} k>205,\\[1mm] 1-(137k^3 \ln k)^{-1}, \text{при} k>e^{628}. \end{array}\right. $$ In particular from this theorems follows that estimation $\gamma1-(137k^3 \ln k)^{-1},$ got by V. A. Plaksin for it is enough greater $k$, remains to be equitable under $\ln k>628$.

Détail

Keywords: The Dirichlet charakter, Dirichlet $L$-function, exceptional set, representation numbers, exceptional zero, exceptional nature, main member, remaining member.