Abstract Generated abstract
Vinogradov announces a new bound for the Riemann zeta function on the line Re s = 1, namely ζ(1+it) = O((log t log log t)^{2/3}) for sufficiently large t. The estimate is derived from a new bound for exponential sums of the form ∑ e^{2πif(x)} with f(x) = −t log x/(2π), under specified size and smoothness conditions, using a development of the author’s earlier method. The resulting zeta estimate implies an improved zero-free region near σ = 1 and corresponding improvements in remainder terms in the prime number theorem, with further applications to broader classes of trigonometric sums to be detailed elsewhere.
Full Text
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