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<r0) 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).