MATHEMATICS

K. A. RODOSSKII

ON A NEW APPLICATION OF I. M. VINOGRADOV’S ESTIMATES TO THE THEORY OF THE RIEMANN ZETA-FUNCTION

(Presented by Academician I. M. Vinogradov on 3 VI 1960)

Let \(N(\Delta,T)\) be the number of zeros of the function \(\zeta(s)\), \(s=\sigma+it\), lying in the rectangle \(\Delta\leqslant \sigma\leqslant 1,\ 0\leqslant t\leqslant T\). If \(\Delta\) and \(T\) are regarded as varying independently of one another, \(T>1\) and \(\Delta\in(1/2,1)\), then all our information on the behavior of \(N(\Delta,T)\) is expressed by inequalities of the form

\[ N(\Delta,T)<c_1\ln^{c_2}T\cdot T^{\varkappa(\Delta)(1-\Delta)}. \tag{1} \]

Here and below \(c_\nu\) denote absolute positive constants, \(\varkappa(\Delta)\geqslant 0\). Various methods for obtaining inequalities of type (1) give different estimates for the function \(\varkappa(\Delta)\). It is known, for example, that

\[ \varkappa(\Delta)\leqslant \min\left\{\frac{3}{2-\Delta},\ 2(1+2c)\right\}, \tag{2} \]

where

\[ c=\lim_{t\to\infty}\frac{\ln|\zeta(1/2+it)|}{\ln t} \]

(see \((^{1})\), Ch. IX, §§ 18—19). The best known value for \(c\) is \(15/92\) (see \((^{1})\), Ch. V, § 16). Inequality (2) is used in the solution of certain problems connected with prime numbers (see \((^{2,3})\)). If it were known that \(\varkappa(\Delta)\leqslant 2\), then in the solution of certain number-theoretic problems one would obtain, qualitatively, the same results as follow from the assumption of the truth of the Riemann hypothesis on the zeros of the zeta-function. However, from (2) one obtains only that \(\lim \varkappa(\Delta)=2+0\) as \(\Delta\to 1/2\). In the author’s paper \((^{4})\) it was pointed out that the estimate of \(\varkappa(\Delta)\) as \(\Delta\to 1\) can be considerably improved by means of the method of trigonometric sums. Later P. Turán \((^{5})\) obtained the estimate

\[ \varkappa(\Delta)\leqslant 2+600(1-\Delta)^{0.01} \tag{3} \]

for \(\Delta\) sufficiently close to 1 (in a note he gives, without proof, the better estimate \(\varkappa(\Delta)\leqslant 2+(1-\Delta)^{0.14}\)).

In this note it is shown that a proper application of new results of I. M. Vinogradov (see \((^{6})\)) leads to the estimate

\[ \varkappa(\Delta)\leqslant 2+c_3(1-\Delta)^{1/3}. \tag{4} \]

The proof of inequality (4) is based on the following lemmas. Below \(\rho=\beta+i\gamma\) denotes a zero of the zeta-function lying in the rectangle \(R\{\Delta\leqslant \sigma\leqslant 1,\ \tfrac12 T\leqslant t\leqslant T\}\).

Lemma 1.
\[ \left|\sum_{n<x} n^{-\rho}\right|\leqslant c_4 x^{1-\beta}T^{-1}\quad \text{for } x\geqslant T. \]

Proof see \((^{1})\), Chapter IV, § 3.

Lemma 2. For \(k\geqslant 7\) and under the condition
\((k+1)^{-1}\ln T \leqslant \ln x \leqslant k^{-1}\ln T\), we have
\[ \left|\sum_{x<n\leqslant T} n^{-\rho}\right| \leqslant c_{5}\ln T\cdot x^{1-c_{6}k^{-2}-\beta}. \]

Proof. This is a simple consequence of I. M. Vinogradov’s estimates of trigonometric sums \((^{6})\) and of Korput’s estimates (see \((^{1})\), Ch. V, Th. 11).

Let
\[ f(s)=\sum_{y<n\leqslant z} a_n n^{-s}, \quad\text{where}\quad a_n=\sum_{d\mid n,\ d>y}\mu(d), \]
and \(\mu(d)\) is the Möbius function.

Lemma 3. If for every zero \(\rho\in R\) of the zeta-function the inequality \(|f(\rho)|\geqslant 1/2\) holds, then the number \(Q(\Delta,T)\) of zeros belonging to \(R\) does not exceed
\[ c_{7}\ln^{8}T\cdot\bigl(Ty^{1-2\Delta}+z^{2-2\Delta}\bigr). \]

Proof follows trivially from Theorem 1 of paper \((^{7})\).

On the basis of the lemmas stated above, the proof of the theorem reduces to the following. Put \(y_1=zT^{-1}\), \(z>T>y>y_1\), and use the identity
\[ \begin{aligned} 1={}& \sum_{n\leqslant y_1}\mu(n)n^{-\rho}\sum_{kn\leqslant z}k^{-\rho} +\sum_{y_1<n\leqslant y}\mu(n)n^{-\rho}\sum_{k\leqslant T}k^{-\rho} \\ &-\sum_{y_1<n\leqslant y}\mu(n)n^{-\rho}\sum_{zn^{-1}<k\leqslant T}k^{-\rho} +\sum_{y<n\leqslant z}\mu(n)n^{-\rho}\sum_{kn\leqslant z}k^{-\rho}. \end{aligned} \tag{5} \]

With the help of Lemma 1 we estimate
\[ \left|\sum_{n\leqslant y_1}\mu(n)n^{-\rho}\sum_{kn\leqslant z}k^{-\rho}\right| + \left|\sum_{y_1<n\leqslant y}\mu(n)n^{-\rho}\sum_{k\leqslant T}k^{-\rho}\right| <\frac14 \tag{6} \]
under the conditions \(z<T^2\), \(\Delta>3/4\), and \(T>T_0\). It is enough to consider only
\[ \Delta \leqslant 1-c_{9}\ln\ln T\cdot \ln^{-1}T. \]
Define \(k\) from the inequalities
\[ \frac{c_6}{4}(k+1)^{-3}\leqslant 1-\Delta \leqslant \frac{c_6}{4}k^{-3} \]
(this is possible for values of \(\Delta\) sufficiently close to 1), and put
\[ y=zT^{-\frac{1}{k+1}}. \]
With the help of Lemma 2 we obtain
\[ \left| \sum_{y_1<n\leqslant y}\mu(n)n^{-\rho} \sum_{zn^{-1}<k\leqslant T} k^{-\rho} \right| \leqslant c_7c_6^{-1}k^2\ln T\cdot z^{1-\Delta}T^{-c_6k^{-3}} <\frac14 \tag{7} \]
for sufficiently large \(T\). From identity (5) and inequalities (6) and (7) we obtain \(|f(\rho)|\geqslant 1/2\), and Lemma 3 can be applied. To estimate \(\chi(\Delta)\), it is enough to consider the expression
\[ Ty^{1-2\Delta}+z^{2-2\Delta}. \]
Its minimal value is obtained from the condition
\[ Ty^{1-2\Delta}=z^{2-2\Delta} \]
or
\[ Tz^{1-2\Delta}T^{\frac{2\Delta-1}{k+1}}=z^{2-2\Delta}, \]
whence it follows that
\[ z=T^{1+\frac{2\Delta-1}{k+1}}, \]
and the estimate
\[ \chi(\Delta)\leqslant 2\left(1+\frac{2\Delta-1}{k+1}\right) \leqslant 2+2\left[\frac{c_6}{4}(1-\Delta)\right]^{1/3}. \]
Putting \((2c_6)^{1/3}=c_3\), we obtain estimate (4).

Voronezh State University

Received
21 V 1960

REFERENCES

1. E. K. Titchmarsh, The Theory of the Riemann Zeta-Function, IL, 1953.
2. N. G. Chudakov, K. A. Rodosskii, Uspekhi Mat. Nauk, 4, 21 (1949).
3. Yu. V. Linnik, Izv. AN SSSR, ser. matem., No. 16, 503 (1952).
4. K. A. Rodosskii, DAN, 86, No. 6, 1069 (1952).
5. P. Turan, Acta Math. Acad. Sci. Hungaricae, 5, fasc. 3–4, 145 (1954).
6. I. M. Vinogradov, Izv. AN SSSR, ser. matem., 22, 161 (1958).
7. K. A. Rodosskii, Izv. AN SSSR, ser. matem., 19, 97 (1955).