On the Function $\zeta(s)$

Academician I. M. Vinogradov

I have found a new estimate for $\zeta(1+it)$:

[
\zeta(1+it)=O\left((\ln t\,\ln\ln t)^{2/3}\right)
\tag{1}
]

$(t\geqslant t_0$, where $t_0$ is a sufficiently large constant $>1)$.
This estimate is a consequence of a new estimate for the sum

[
S=\sum_{a<x\leq b} e^{2\pi i f(x)};\qquad f(x)=-\frac{t\ln x}{2\pi},
]

where, for an integer $n\geqslant 7$, the conditions

[
a<b\leqslant 2a,\qquad t=a^{\,n-\theta},\qquad 0<\theta\leqslant 1
]

are satisfied.

Namely, for $a>(4n)^{16n^2}$, for the sum $S$ I obtained an estimate which can be put in the form (the letters $c,c_1,c_2,\ldots$ denote absolute positive constants)

[
|S|\leqslant e^{c\ln^2 n} a^{1-\frac{c_1}{n^2\ln n}}.
]

Estimate (1), and other estimates analogous to it obtained by the same method, entail an improvement of a number of generally known results in the theory of the distribution of prime numbers. For example: $\zeta(s)=\zeta(\sigma+it)$ has no zeros in the region

[
\sigma\geqslant 1-\frac{c_2}{(\ln t\,\ln\ln t)^{2/3}}.
]

Correspondingly, the remainder term in the asymptotic formula for the number $\pi(N)$ of primes not exceeding $N$ is also improved, and so on. The method I have applied to derive the estimate for the sum $S$ is a certain development of my earlier method. It can also be successfully applied to the derivation of new estimates for trigonometric sums belonging to broader classes. A detailed exposition of my new results will be given elsewhere.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
31 X 1957