MATHEMATICS

N. M. KOROBOV

ON THE ZEROS OF THE FUNCTION \(\zeta(s)\)

(Presented by Academician I. M. Vinogradov, 27 VIII 1957)

In the present paper new estimates of rational trigonometric sums are given, and applications of these estimates to the theory of the Riemann zeta-function and to the question of the distribution of prime numbers are presented. For the zeta-function, a refinement of the estimate of \(|\zeta(s)|\) and an improvement of the boundary of the real part of the zeros of \(\zeta(s)\) are obtained. In the question of the distribution of prime numbers, the corresponding improvement of the estimate of the modulus of the difference \(\pi(x)-\operatorname{li} x\) is given.

Let \(q, a_1,\ldots,a_{n+1}\) be integers, \((q,a_{n+1})=1\), and \(1<n<p_1-1\), where \(p_1\) is the least prime divisor of \(q\). Then the following theorem is valid.

Theorem 1. Whatever fixed \(\varepsilon\) may be \((0<\varepsilon\le 0.5)\), there exist an absolute constant \(C\) and a constant \(\alpha=\alpha(\varepsilon)\) such that, for \(P=q^{1/r}\) on the interval \(n+\varepsilon\le r\le n+1-\varepsilon\), the estimate

\[ \left| \sum_{x=1}^{P} e^{2\pi i \frac{a_1x+\cdots+a_{n+1}x^{n+1}}{q}} \right| < CP^{\,1-\frac{\alpha}{(n\ln n)^{2.5}}}. \tag{1} \]

The estimate (1) on the interval \(n+\varepsilon\le r\le n+1-\varepsilon\) represents a strengthening of the result indicated in my paper \((^1)\). The proof of the theorem, as in \((^1)\), is based on a combination of I. M. Vinogradov’s method with a new approach to estimates of trigonometric sums, in which the rationality of the sums under consideration is substantially used.

Theorem 2. As \(|t|\to\infty\), for every fixed \(\varepsilon>0\) the estimate

\[ \zeta(1+it)=O\{(\ln |t|)^{5/7+\varepsilon}\} \]

is valid.

The proof of the theorem is based on reducing the question of estimating sums of the form

\[ \sum_{x=Q+1}^{Q+P} x^{ti} \]

to estimates of rational trigonometric sums satisfying the conditions of Theorem 1.

From Theorem 2, in the usual way \((^2,^3)\), we obtain the following assertions:

Theorem 3. For every fixed \(\varepsilon>0\) there exists a positive constant \(A=A(\varepsilon)\) such that, in the region

\[ \sigma \ge 1-\frac{A}{(\ln |t|)^{5/7+\varepsilon}}, \]

the function \(\zeta(\sigma+it)\) has no zeros.

Theorem 4. For every fixed \(\varepsilon>0\) there exists a positive-

constant \(a=a(\varepsilon)\) such that, as \(x\to\infty\), the equality

\[ \pi(x)=\operatorname{li}x+O\left(xe^{-a\{\ln x\}^{7/12-\varepsilon}}\right). \]

holds.

Remark. Using the results of Theorem 1, one can obtain an analogous refinement of estimates in a number of other questions, in particular in questions concerning the bound for the real part of the zeros of Dirichlet \(L\)-functions and the remainder term in the formula for the number of primes \(p\le x\) belonging to a given arithmetic progression.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
26 VIII 1957

REFERENCES

1. N. M. Korobov, DAN, 118, No. 2 (1958).
2. E. K. Titchmarsh, The Theory of the Riemann Zeta-Function, IL, 1953.
3. A. E. Ingham, The Distribution of Prime Numbers, M.—L., 1936.