MATHEMATICS

A. A. KARATSUBA

ESTIMATES OF TRIGONOMETRIC SUMS OF A SPECIAL TYPE AND THEIR APPLICATIONS

(Presented by Academician I. M. Vinogradov, 29 X 1960)

Consider the trigonometric sum

\[ S=\sum_{x=1}^{N} e^{2\pi i\left(\frac{a_1x}{p^n}+\frac{a_2x^2}{p^{n-1}}+\cdots+\frac{a_nx^n}{p}\right)}, \tag{1} \]

where \((a_\nu,p)=1,\ \nu=1,2,\ldots,n\).

Theorem 1. Let \(S\) be a sum of the form (1), \(p\leq N\leq p^n\), \(\log p\gg n^2\log^3 n\). Then the inequality

\[ |S|\leq c_1N^{1-\frac{c_2}{n^2}}, \]

holds, where \(c_1,c_2\) are absolute constants.

In the proof of this theorem one uses the theorem on the mean of I. M. Vinogradov \((^1)\), Lemma 1 of the paper \((^2)\), and the special form of the sum (1) (we make essential use of several coefficients of the polynomial

\[ f(x)=\frac{a_1x}{p^n}+\frac{a_2x^2}{p^{n-1}}+\cdots+\frac{a_nx^n}{p}. \]

Trigonometric sums of the form (1) occur in the theory of Dirichlet \(L\)-series and in certain other number-theoretic questions.

From Theorem 1 and results of A. G. Postnikov \((^3)\) on Dirichlet \(L\)-series, the following theorem follows easily:

Theorem 2. Let \(\chi(k)\) be a primitive character modulo \(D=p^n\), \(p\) a prime \(>2\), and \(\log p\gg n^2\log^3 n\).

Then, denoting by \(S_N\) the sum \(\sum_{k=1}^{N}\chi(k)\), we have

\[ |S_N|\leq \begin{cases} p^2, & \text{if } N\leq p^2,\\ c_3N^{1-\frac{c_4}{n^2}}, & \text{if } p^2\leq N\leq p^n, \end{cases} \]

where \(c_3,c_4\) are absolute constants.

Further, in the usual way we obtain the theorem:

Theorem 3. Let \(n^2\log^3 n\ll \log p\leq n^\theta\), \(\chi(k)\) be a primitive character modulo \(D=p^n\), \(p\) a prime \(>2\). Then \(L(s,\chi)\) has no zeros in the region

\[ |s|<c_5,\qquad \sigma>1-\frac{1}{\log^{\theta+1}D}, \]

where \(c_5\) is a constant.

Corollary. Let \(\varepsilon\) be an arbitrarily small positive quantity and let
\[ n^2 \log^3 n \ll \log p \ll n^{2+\varepsilon}. \]
Then \(L(s,\chi)\) has no zeros in the domain
\[ |s|<c_5,\qquad \sigma>1-\frac{1}{\log^{1/3+\varepsilon} D}. \]

The best result previously obtained in the theory of zeros of Dirichlet \(L\)-series is the following (see \((^4)\)): if \(\varepsilon\) is an arbitrarily small positive quantity and
\[ \log p \ll \frac{n^\varepsilon}{\log^{3/4} n}, \]
then \(L(s,\chi)\) has no zeros in the domain
\[ |s|<c_5,\qquad \sigma>1-\frac{1}{\log^{3/4+\varepsilon} D}. \]
Theorem 2 also improves the corresponding result of \((^3)\).

I express my gratitude to my adviser N. M. Korobov for the great assistance he gave me in carrying out this work.

Moscow State University
named after M. V. Lomonosov

Received
27 X 1960

REFERENCES

\(^1\) I. M. Vinogradov, The method of trigonometric sums in number theory, Publishing House of the USSR Academy of Sciences, 1947.
\(^2\) N. M. Korobov, UMN, 13, no. 4 (82) (1958).
\(^3\) A. G. Postnikov, Izv. AN SSSR, ser. matem., 19, 11 (1955).
\(^4\) S. M. Rozin, Izv. AN SSSR, ser. matem., 23, 503 (1959).