N. M. KOROBOV

ESTIMATES OF WEYL SUMS AND THE DISTRIBUTION OF PRIME NUMBERS

(Presented by Academician I. M. Vinogradov on 17 V 1958)

The present paper is a continuation of the papers \((^{1-3})\) and strengthens the results obtained in them.

Denote by \(N_k^{(\nu)}(\lambda_1,\ldots,\lambda_n)\) the number of solutions of the system of equations

\[ x_1^\nu+\cdots+x_k^\nu = y_1^\nu+\cdots+y_k^\nu+\lambda_\nu, \qquad 1\leq x,y\leq P \quad (\nu=1,2,\ldots,n). \tag{1} \]

It is easy to show that for the trigonometric sum

\[ S(\omega_1,\ldots,\omega_n) = \sum_{x=1}^{P} e^{2\pi i(\omega_1x+\cdots+\omega_nx^n)} \]

for any integer \(k\geq 1\) the equality

\[ |S(\omega_1,\ldots,\omega_n)|^{2k} = \sum_{\lambda_1,\ldots,\lambda_n} N_k^{(P)}(\lambda_1,\ldots,\lambda_n) e^{2\pi i(\omega_1\lambda_1+\cdots+\omega_n\lambda_n)} \tag{2} \]

holds, where the summation is over all \(\lambda_\nu\) \((\nu=1,2,\ldots,n)\) satisfying the condition \(|\lambda_\nu|<kP^\nu\). Further, from the definition of the quantities \(N_k^{(P)}(\lambda_1,\ldots,\lambda_n)\) we obtain without difficulty the relations

\[ \sum_{\lambda_{s+1},\ldots,\lambda_n} N_k^{(P)}(\lambda_1,\ldots,\lambda_n) = N_k^{(P)}(\lambda_1,\ldots,\lambda_s). \tag{3} \]

Consider the Weyl sum \(S=S(\alpha_1,\ldots,\alpha_{n+1})\).

Fundamental lemma. Let \(P_1\leq P\), \(\beta_\nu=C_{\nu+1}^{1}\alpha_{\nu+1}\lambda_1+\cdots+C_{n+1}^{\,n+1-\nu}\times \alpha_{n+1}\lambda_{n+1-\nu}\) \((\nu=1,2,\ldots,n)\), and

\[ V_{kk_1} = \sum_{\lambda_1,\ldots,\mu_n} N_k^{(P)}(\lambda_1,\ldots,\lambda_n) N_{k_1}^{(P_1)}(\mu_1,\ldots,\mu_n) e^{2\pi i(\beta_1\mu_1+\cdots+\beta_n\mu_n)}, \]

where the summation is over all \(|\lambda_\nu|<kP^\nu\), \(|\mu_\nu|<k_1P_1^\nu\). Then

\[ \left|\frac{1}{2}S\right|^{4kk_1} \leq P^{2k(2k_1-1)}P_1^{-2k}V_{kk_1} + (2P_1)^{4kk_1}. \]

The proof of the lemma is obtained with the aid of Hölder’s inequality and the relations (2), (3), from the estimate

\[ |S| \leq \frac{1}{P_1} \sum_{y=1}^{P_1} \left| \sum_{x=1}^{P} e^{2\pi i f(x+y)} \right| + 2P_1, \]

where \(f(x)=\alpha_1x+\cdots+\alpha_{n+1}x^{n+1}\).

Theorem 1. Let
\[ \alpha_{n+1}=\frac{a}{q}+\frac{\theta}{q^2},\quad (a,q)=1,\quad |\theta|<1,\quad q=P^r; \]
let \(r\) belong to the interval
\[ \sqrt n\ln n<r<n-\sqrt n\ln n. \]
Then there exist absolute constants \(C\) and \(\gamma>0\) such that
\[ \left|\sum_{x=1}^{P} e^{2\pi i(\alpha_1x+\ldots+\alpha_{n+1}x^{n+1})}\right| \ll CP^{\,1-\frac{\gamma}{n^2\ln n}}. \]

Proof. Write \(V_{kk_1}\) in the form
\[ V_{kk_1}= \sum_{\mu_1,\ldots,\mu_n} N_{k_1}^{(P_1)}(\mu_1,\ldots,\mu_n) \sum_{\lambda_1,\ldots,\lambda_n} N_k^{(P)}(\lambda_1,\ldots,\lambda_n) e^{2\pi i(\beta_1\mu_1+\ldots+\beta_n\mu_n)}. \tag{4} \]
According to the definition of the quantities \(\beta_\nu\), the expression
\(\beta_1\mu_1+\ldots+\beta_n\mu_n\) is a linear homogeneous function of the quantities
\(\lambda_1,\ldots,\lambda_n\), and, consequently, by virtue of (2) the inner sum in (4) is nonnegative. Hence, putting
\[ N_{k,n}^{(P)}=\max_{\lambda_1,\ldots,\lambda_n}N_k^{(P)}(\lambda_1,\ldots,\lambda_n), \]
we obtain*
\[ V_{kk_1}\ll N_{k_1,n}^{(P_1)} \sum_{\mu_1,\ldots,\mu_n}\sum_{\lambda_1,\ldots,\lambda_n} N_k^{(P)}(\lambda_1,\ldots,\lambda_n) e^{2\pi i(\beta_1\mu_1+\ldots+\beta_n\mu_n)} \ll \]
\[ \ll N_{k_1,n}^{(P_1)} \sum_{\lambda_1,\ldots,\lambda_n} N_k^{(P)}(\lambda_1,\ldots,\lambda_n) \min\left(2k_1P_1,\frac1{(\beta_1)}\right)\cdots \min\left(2k_1P_1^n,\frac1{(\beta_n)}\right). \]

Choose
\[ s=\min(r,n-r). \]
Applying, for \(\nu\le n-s\), the estimate
\[ \min\left(2k_1P_1^\nu,\frac1{(\beta_\nu)}\right)\ll 2k_1P_1^\nu, \]
by virtue of (3), we obtain from this
\[ V_{kk_1}\ll (2k_1)^{\,n-s}P_1^{\frac{(n-s)(n-s+1)}2}\,N_{k_1,n}^{(P_1)}V_k', \]
where
\[ V_k'=\sum_{\lambda_1,\ldots,\lambda_s}N_k^{(P)}(\lambda_1,\ldots,\lambda_s) \min\left(2k_1P_1^n,\frac1{(\beta_n)}\right)\cdots \min\left(2k_1P_1^{\,n-s+1},\frac1{(\beta_{n-s+1})}\right). \]

Further, using the estimate
\[ \sum_{x=Q+1}^{Q+T}\min\left(U,\frac1{(mx+\beta)}\right) \ll C_0\left(\frac{mT}{q}+1\right)(U+q\ln q), \]
where
\[ \alpha=\frac{a}{q}+\frac{\theta}{q^2},\quad (a,q)=1,\quad |\theta|<1 \]
and \(C_0\) is an absolute constant, we obtain
\[ V_k'\ll N_{k,s}^{(P)} \sum_{\lambda_1,\ldots,\lambda_s} \min\left(2k_1P_1^n,\frac1{(\beta_n)}\right)\cdots \min\left(2k_1P_1^{\,n-s+1},\frac1{(\beta_{n-s+1})}\right) \ll \]
\[ \ll (2k_1C_0)^s2^{sn}P_1^{\frac{n(n+1)}2-\frac{(n-s)(n-s+1)}2}N_{k,s}^{(P)}; \]
\[ V_{kk_1}\ll (2k_1C_0)^n2^{sn}P_1^{\frac{n(n+1)}2}N_{k_1,n}^{(P_1)}N_{k,s}^{(P)}. \tag{5} \]

\[ \text{* By }(\beta)\text{ is denoted the distance from }\beta\text{ to the nearest integer.} \]

Choose \(k_1=[M_1 n^2\ln n]\) and \(k=[Ms^2]\), where \(M_1\) and \(M\) are sufficiently large positive constants. Then, by I. M. Vinogradov’s mean value theorem \((4,^5)\), we obtain

\[ N_{k_1,n}^{(P_1)} \ll e^{c_1 n^3\ln^3 n} P_1^{\,2k_1-\frac{n(n+1)}{2}+\frac12}, \qquad N_{k,s}^{(P)} \ll e^{c_1 s^3\ln s} P^{\,2k-\frac{s^2}{4}}, \]

where \(c_1\) is an absolute constant. Hence, by virtue of (5), applying the main lemma and observing that \(s>\sqrt n\ln n\), we obtain the assertion of the theorem without difficulty.

Let, as above,

\[ \alpha_{n+1}=\frac{a}{q}+\frac{\theta}{q^2},\qquad (a,q)=1,\qquad |\theta|<1,\qquad q=P^r . \]

The estimate of Theorem 1 can be strengthened if one restricts oneself to a somewhat less wide interval of variation of \(r\).

Theorem 2. Whatever fixed \(\varepsilon>0\) may be, there exist an absolute constant \(C\) and a constant \(\gamma=\gamma(\varepsilon)\) such that, for \(\varepsilon n<r<n-\varepsilon n\), the estimate

\[ \left| \sum_{x=1}^{P} e^{2\pi i(\alpha_1 x+\cdots+\alpha_{n+1}x^{n+1})} \right| \ll C P^{\,1-\frac{\gamma}{n^2}} . \tag{6} \]

The proof of Theorem 2 differs from the preceding proof only in that, instead of \(k_1=[M_1n^2\ln n]\), one should choose \(k_1=[M_1n^2]\). Estimates of the form (6) were obtained by me earlier for the case of rational trigonometric sums \((^1)\). In Theorem 2 these estimates are extended to the case of arbitrary Weyl sums.

Consider the polynomial

\[ f(x)=\alpha_1x+\cdots+\alpha_{n+1}x^{n+1}, \]

some of whose coefficients are rational:

\[ \alpha_\nu=\frac{a_\nu}{q},\qquad \nu=s+2,\ s+3,\ldots,3s,\qquad 1\le s\le \frac{n+1}{3}. \]

Denote by \(\Delta_s\) the determinant of order \(s\),
\[ \Delta_s=\left|C_{s+i+j}^{\,i} a_{s+i+j}\right|. \]

Theorem 3. Let \(\delta\) be an arbitrary fixed number from the interval \(0<\delta<1/3\), \(n\delta\le s\le \dfrac{n+1}{3}\), \(s+1\le r\le 2s(1-\delta)\), \(q=P^r\), and \((\Delta_s,q)=1\). Then there exist constants \(C=C(\delta)\) and \(\gamma=\gamma(\delta)\) such that

\[ \left| \sum_{x=1}^{P} e^{2\pi i(\alpha_1x+\cdots+\alpha_{n+1}x^{n+1})} \right| < C P^{\,1-\frac{\gamma}{n^2}} . \]

The proof of the theorem is based on the main lemma and certain additional considerations concerning the quantities \(N_k^{(P)}(\lambda_1,\ldots,\lambda_n)\), which make it possible to estimate more sharply the sum \(V_{kk}\) from the main lemma. Theorem 3 has various applications. Thus, for example, with its aid the following assertions are obtained:

Theorem 4. As \(|t|\to\infty\), for the Riemann function \(\zeta(s)\) the estimate

\[ \zeta(1+it)=O(\ln^{2/3}|t|). \tag{7} \]

Theorem 5. Let \(\pi(x)\) be the number of primes not exceeding \(x\). There exists a constant \(a>0\) such that the estimate

\[ \pi(x)-\int_{2}^{x}\frac{du}{\ln u}=O\left(xe^{-a\ln^{3/5}x}\right). \tag{8} \]

holds.

Theorems 4 and 5 strengthen the assertions of the papers \((^{2,6})\). An analogous strengthening of the results is also obtained in the questions mentioned in \((^3)\).

Proof correction note. After the present note had been submitted for publication, a paper by I. M. Vinogradov \((^7)\) appeared, in which estimates (7) and (8) were also obtained. The estimates in \((^7)\) are based on an inequality whose idea coincides with the idea of the inequalities first applied in the papers \((^{1,2})\) (see also \((^{8,9})\)).

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

Received
5 V 1958

CITED LITERATURE

\(^1\) N. M. Korobov, DAN, 118, No. 2, 431 (1958).
\(^2\) N. M. Korobov, DAN, 118, No. 3, 231 (1958).
\(^3\) N. M. Korobov, DAN, 119, No. 3, 433 (1958).
\(^4\) I. M. Vinogradov, Izv. AN SSSR, Ser. Mat., 15, 109 (1951).
\(^5\) Hua Loo-Keng, Quart. J. Math., 20, 48 (1949).
\(^6\) I. M. Vinogradov, DAN, 118, No. 4, 631 (1958).
\(^7\) I. M. Vinogradov, Izv. AN SSSR, Ser. Mat., 22, 161 (1958).
\(^8\) N. M. Korobov, Uspekhi Mat. Nauk, 13, No. 2 (80), 243 (1958).
\(^9\) N. M. Korobov, Uspekhi Mat. Nauk, 13, No. 4 (82), 185 (1958).