Vinogradov's Integral and Bounds for the Riemann Zeta Function

The main result is an upper bound for the Riemann zeta functionin the critical strip: with A = 76.2 and B = 4.45, valid for 1 and |t| 3. The previousbest constant B was 18.5. Tools include a variant of the Korobov–Vinogradovmethod of bounding exponential sums, an explicit version ofT. D. Wooley's bounds for Vinogradov's integral, and explicitbounds for mean values of exponential sums over numbers withoutsmall prime factors, also using methods of Wooley. An auxiliaryresult is the exponential sum bound , where N is a positive integer, t is a real number, = log (t)/(logN) and 2000 Mathematical Subject Classification: primary 11M06, 11N05,11L15; secondary 11D72, 11M35.