Estimates of short cubic double exponential sums with a long continuous summation

Z. Kh. Rakhmonov ; F. Z. Rakhmonov ; B. M. Zamonov

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 217-231

Résumé

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 \begin{align*} (\alpha ;x,y) = \sum_{x-y\le x} \Lambda(n) e(\alpha n^k),\quad \alpha=\frac{a}{q}+\lambda,\quad |\lambda|\le \frac{1}{q\tau},\quad 1\le q\le \tau \end{align*} when $$ \exp(c(\ln \ln x)^2)\ll q \ll x^{1/3},\qquad y>x^{2/3+\varepsilon}, $$ This estimate is based on “Vinogradov sieve” and for $k=1$ utilizes estimates of short double exponential sums of the form $$ J_k(\alpha;x,y,M,N)=\sum_{M\le 2M}a(m)\sum_{U\le 2N \atop x-y\le x}b(n)e(\alpha (mn)^k), $$ where $a(m)$ and $b(n)$ are arbitrary complex-valued functions, $M$, $N$ are positive integers, $N\le U2N$, $x>x_0$, $y$ are real numbers. Later, B. Haselgrove, V. Statulyavichus, Pan Cheng-Dong and Pan Cheng-Biao, Zhan Tao obtained a nontrivial estimate for the sum $S_1(\alpha;x,y)$, $y\ge x^{\theta}$, where $q$ was an arbitrary integer, and successfully proved an asymptotic formula for ternary Goldbach problem with almost equal summands satisfying $|p_i-N/3|\le H$, $ H=N^{\theta}$, respectively when $$ \theta=\frac{63}{64}+\varepsilon, \qquad \frac{279}{308}+\varepsilon, \qquad \frac{2}{3}+\varepsilon ,\qquad \frac{5}{8}+\varepsilon. $$ J. Liu and Zhan Tao studied the sum $J_2(\alpha;x,y,M,N)$ and obtained a non-trivial estimate for the sum $S_2(\alpha ;x,y)$ when $y\ge x^{\frac{11}{16}+\varepsilon}$. This paper is devoted to obtaining non-trivial estimates for the sum $J_3(\alpha;x,y,M,N)$, with a “long” continuous summation over minor arcs. Bibliography: 12 titles.

Détail

Keywords: Short double exponential sums, nontrivial estimate, estimation method for short exponential sums over primes.