VINOGRADOV'S INTEGRAL AND BOUNDS FOR THE RIEMANN ZETA FUNCTION

Abstract

The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid for $\frac12 \le \sigma \le 1$ and $|t| \ge 3$. The previous best constant $B$ was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound $S(N, t) \le 9.463 N^{1 - 1/(133.66\lambda^2)}$, where $N$ is a positive integer, $t$ is a real number, $\lambda = (\log t)/(\log N)$ and$S(N,t) = \max_{0 < u \le 1} \max_{N < R \le 2N} \left| \sum_{N < n \le R} (n + u)^{-it} \right|.$$2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.