Abstract
I. M. Vinogradov pioneered the study of short exponential sums with primes. For k = 1 using his method of estimating sums with primes, he obtained a non-trivial estimate for sums of the form Sk(α; x, y) = X x−y x2/3+e, This estimate is based on “Vinogradov sieve” and for k = 1 utilizes estimates of short double exponential sums of the form Jk(α; x, y,M,N) = X M x0, y are real numbers. Later, B. Haselgrove, V. Statulyavichus, Pan Cheng-Dong and Pan Cheng-Biao, Zhan Tao obtained a nontrivial estimate for the sum S1(α; x, y), y > xθ, where q was an arbitrary integer, and successfully proved an asymptotic formula for ternary Goldbach problem with almost equal summands satisfying |pi − N/3| 6 H, H = Nθ, respectively when θ = 63 64 + e, 279 308 + e, 2 3 + e, 5 8 + e. J. Liu and Zhan Tao studied the sum J2(α; x, y,M,N) and obtained a non-trivial estimate for the sum S2(α; x, y) when y > x 11 16+e. This paper is devoted to obtaining non-trivial estimates for the sum J3(α; x, y,M,N), with a “long” continuous summation over minor arcs.
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