Abstract
Full Text
UDC 511
MATHEMATICS
A. A. KARATSUBA
ON SUMS OF CHARACTERS WITH PRIME NUMBERS
(Presented by Academician I. M. Vinogradov on 27 X 1969)
Notation: (q) is a sufficiently large prime number; (\chi) is a nonprincipal character mod (q); (p) is a prime number; (k) is an arbitrary integer not divisible by (q); (\varepsilon>0) is arbitrarily small and not always the same.
In the paper the sums (S_N) are studied,
[
S_N=\sum_{p\leq N}\chi(p+k).
\tag{1}
]
Such sums were first studied by I. M. Vinogradov (see ((^{1-5}))) with the aid of the method he created in 1934–1937 for estimating trigonometric sums, which made it possible to obtain a number of fundamental results in number theory.
The strongest result ((^5)) is formulated as follows:
[
|S_N|\ll N^{1+\varepsilon}\Delta,
\tag{2}
]
where (\Delta=q^{1/4}N^{-1/3}+N^{-1/10}).
It is clear from (2) that the estimate will already be nontrivial for (N>q^{3/4+\varepsilon}). We note that, if one assumes the generalized Riemann hypothesis, then from it one can obtain a nontrivial estimate of (1) only for (N>q^{1+\varepsilon}). In the present work, by combining the method of I. M. Vinogradov with ideas from the author’s paper ([6]), a new estimate is obtained, nontrivial already for (N>q^{1/2+\varepsilon}).
Theorem 1. Let (\omega) be an arbitrary number in the interval (0<\omega\leq 1/4), and let (q^{1/2+\omega}\leq N\leq q). Then the estimate
[
|S_N|\ll Nq^{-\gamma\omega^2},
]
holds, where (\gamma>0) is an absolute constant, and the constant in (\ll) depends only on (\omega).
We shall formulate three main lemmas needed for the proof of the theorem.
Lemma 1. Let (\gamma_1) and (\gamma_2) be positive numbers,
[
q^{\gamma_1}\leq M<M_1\leq 2Mq^{1/2+\gamma_2},
]
[
T=\sum_{U<u\leq U}^{\prime}\left|\sum_{M<m\leq \min(M_1,Nu^{-1})}^{\prime}\chi(m+ku^{-1})\right|,
]
where (u,m) run through values that are products, respectively, of (c_1) and (c_2) factors, each of which, independently of the others, runs through its own increasing sequence of positive integers. Then
[
T\ll UMq^\varepsilon\Delta,
]
where (\Delta=q^{0.5\gamma_1\gamma_2}+q^{-0.25\gamma_1}).
Lemma 2. Let (\gamma_1,\gamma_2,\gamma_3) be positive numbers,
[
0<\gamma<0.5(0.5\gamma_3-\gamma_2).
]
[
q^{\gamma_1} \ll M \ll q^{1/4+\gamma_2}, \qquad
M < M_1 \leq 2M, \qquad
q^{1/2+\gamma_3} \ll UM \leq q,
]
[
T=\sum_{U<u\leq 2U}^{\prime}\left|
\sum_{M<m\leq \min(M_1,Nu^{-1})}\chi(m+ku^{-1})
\right|,
]
where (u) runs through values that are products of factors each of which, independently of the others, runs through its own increasing sequence of positive integers, and (m) runs through consecutive integer values of the interval ((M,\min(M_1,Nu^{-1})]). Then
[
T\ll UMq^\varepsilon\Delta,
]
where
[
\Delta=q^{-\gamma_1\left(1-\frac{1}{1+0.5\gamma_3-\gamma_2-2\gamma_1}\right)}.
]
Lemma 3. Let (\gamma>0), (M) be an integer,
[
q^{1/4+\gamma}\ll M\ll q^{5/8}, \qquad
0\leq a\leq q-1, \qquad
T=\sum_{m=1}^{M}\chi(m+a).
]
Then
[
T\ll Mq^\varepsilon\Delta,
]
where (\Delta=q^{-0.5\gamma^2}).
The proof of Theorem 1 is carried out by combining the method of I. M. Vinogradov (see, for example, the proof of the theorem in ({}^{3})) with the results of Lemmas 1, 2, 3.
From Theorem 1 one rather easily obtains new theorems on the distribution of residues and nonresidues (\bmod\, q) of arbitrary degree in sequences of the form (p+k,\ p\leq N).
Theorem 2. Let (n\mid(q-1)), (2\leq n\leq q-2), and (0\leq s\leq n-1). Then, under the conditions of Theorem 1, for the number (T_s) of those numbers (p+k) which satisfy the conditions
[
p\leq N, \qquad \operatorname{ind}(p+k)\equiv s \pmod n,
]
the formula
[
T_s=\frac{1}{n}\pi(N)+O(Nq^{-\gamma\omega^2})
]
holds.
Theorem 3. Under the conditions of Theorem 1, the number of quadratic residues and the number of quadratic nonresidues (\bmod\, q) of the form (p+k,\ p\leq N), is equal to
[
\frac{1}{2}\pi(N)+O(Nq^{-\gamma\omega^2}).
]
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Moscow
Received
21 X 1969
REFERENCES
- I. M. Vinogradov, Selected Works, Publishing House of the Academy of Sciences of the USSR, 1952.
- I. M. Vinogradov, Mat. sbornik, 3 (45), 311 (1938).
- I. M. Vinogradov, Izv. AN SSSR, ser. matem., 17, 197 (1952).
- I. M. Vinogradov, Izv. AN SSSR, ser. matem., 17, 285 (1953).
- I. M. Vinogradov, Izv. AN SSSR, ser. matem., 30, No. 3, 481 (1966).
- A. A. Karatsuba, DAN, 180, No. 6, 1287 (1968).