Vinogradov's integral and bounds for the Riemann zeta function

Abstract

We show for all $1/2 \le \sigma \le 1$ and $t\ge 3$ that $\zeta(\sigma+it)| \le 76.2 t^{4.45 (1-\sigma)^{3/2}}$, where $\zeta$ is the Riemann zeta function. This significantly improves the previous bounds, where $4.45$ is replaced by $18.8$. New ingredients include a method of bounding $\zeta(s)$ in terms of bounds for Vinogradov's Integral (aka Vinogradov's Mean Value) together with bounds for "incomplete Vinogradov systems", explicit bounds for Vinogradov's integral which strengthen slightly bounds of Wooley (Mathematika 39 (1992), no. 2, 379-399), and explicit bounds for the count of solutions of "incomplete Vinogradov systems", following ideas of Wooley (J. Reine Angew. Math. 488 (1997), 79-140)