Sum of short exponential sums with prime numbers
Firuz Rakhmonov
TL;DR
This paper extends Vinogradov's method to short exponential sums with prime arguments and a polynomial phase $f$ of degree $n$, under the approximation $\alpha= a/q + \theta/q^2$. The main contribution is a bound for $V_n(K,x,y)=\sum_{k=1}^K\left|\sum_{x-y<p\le x} e(kf(p))\right|$ showing $V_n(K,x,y) \ll Ky \left( \frac{1}{q} + \frac{1}{y} + \frac{q}{K y^n} + \frac{1}{K^{2^{n-1}}} \right)^{2^{-(n+1)}} (\ln x)^{\frac{n^2}{2^{n+1}}}$, which strengthens Vinogradov's estimates and generalizes them to short-prime intervals. The proof combines Weyl's method with Weyl differencing and a detailed treatment of finite differences $\Delta_j$ for polynomials, together with auxiliary lemmas to handle sums over primes via a reduction to linear phases. The results advance understanding of uniform distribution modulo one and nontrivial bounds for short exponential sums over primes, with potential implications for problems on fractional parts and primes in short intervals.
Abstract
For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = αu^n + α_{n-1}u^{n-1} + \ldots + α_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer, $α$ is a real number such that $\left|α- \frac{a}{q}\right| \le \frac{1}{q^2}$, $(a, q) = 1$, $q \ge 1$ and $\mathscr{L} = \ln x$, an estimate of the form $$ \sum_{k=1}^K \left| \sum_{x - y < p \le x} e(kf(p)) \right| \ll K y \left( \frac{1}{q} + \frac{1}{y} + \frac{q}{K y^n} + \frac{1}{K^{2^{n-1}}} \right)^{2^{-n-1}} {\mathscr{L}}^{\frac{n^2}{2^{n+1}}}, $$ is obtained, which represents a strengthening and generalization of the corresponding estimate of I.M.Vinogradov. Keywords: short exponential sum of G.Weyl with prime numbers, uniform distribution modulo one, nontrivial estimate, fractional part. Bibliography: 18 references.