A remark on Vinogradov's mean-value theorem (Q1354695)

I. M. Vinogradov's well-known mean value theorem for exponential integrals plays an important rôle in multiplicative number theory. It is known to furnish the bounds \[ \sum_{n\leq N}n^{it} \ll N\exp\left(-A{\log^3N\over\log^2t}\right) \qquad(A > 0, 1 \ll N \ll t)\leqno(1) \] and \[ \zeta(\sigma + it) \ll t^{B(1-\sigma)^{3/2}}\log^{2/3}t\qquad(B > 0, 1/2 \leq \sigma \leq1, t \geq 3),\leqno(2) \] from which many important consequences follow (the whole topic is extensively discussed e.g. in the reviewer's book [The Riemann zeta-function, Wiley, New York (1985; Zbl 0758.11036)]). The authors give a detailed deduction of (1) and (2) by using Vinogradov's mean value theorem with the explicit value \(B = 75\) (not the best known value).