ESTIMATES OF SHORT CUBIC DOUBLE EXPONENTIAL SUMS WITH A LONG CONTINUOUS SUMMATION

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<n6x Λ(n)e(αnk), α = a q + λ, |λ| 6 1 qτ , 1 6 q 6 τ when exp(c(ln ln x)2) ≪ q ≪ x1/3, y > x2/3+ε, 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<m62M a(m) X U<n62N x−y<mn6x b(n)e(α(mn)k), where a(m) and b(n) are arbitrary complex-valued functions, M, N are positive integers, N 6 U < 2N, x > 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 + ε, 279 308 + ε, 2 3 + ε, 5 8  + ε. 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+ε. 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.