On an equation with prime numbers
It is well-known that in 1937 I. M. Vinogradov setted the Goldbach conjecture for large odd integers, namely, there is a (computable) \(n_0>0\) such that for any odd integer \(n \geq n_0\) the equation \(n=p_1 + p_2+p_3\) is soluble in primes \(p_1, p_2,p_3\). In this paper the authors investigate a variant of the Goldbach-Vinogradov result by considering the equation \[ N= [p^c_1] + [p^c_2] + [p^c_3] \tag{1} \] where \([x]\) denotes the integral part of the real \(x\). They obtain: Let \(c\) be a given constant with \(1<c < 17/16\) and \(N\) be a sufficiently large positive integer. Then for any \(\varepsilon>0\), \[ \sum \prod^3_{j=1} \log p_j = \biggl\{ \Gamma^3 \bigl((1/c)+1 \bigr)/ \Gamma (3/c) \biggr\} N^{(3/c)-1} + O\biggl( N^{(3/c)-1} \exp \bigl( -(\log N)^{(1/3)- \varepsilon} \bigr) \biggr), \] where the summation \(\sum\) is over all primes \(p_1, p_2, p_3\) satisfying (1) and \(\Gamma\) denotes the Gamma function. This result is parallel to the second author's work [Acta Arith. 61, 289-306 (1992; Zbl 0762.11033)] on the inequality, \(|p^d_1 + p^d_2 + p^d_3-N |<N^{1-(15/(14d))} \log^9N\) where \(1<d<15/14\).
scientific article; zbMATH DE number 898900