Abstract
Full Text
UDC 511.9
A. A. KARATSUBA
ON TRIGONOMETRIC SUMS
(Presented by Academician I. M. Vinogradov, 21 III 1969)
In the paper [2] Hua Loo-keng proved:
If (n \geqslant 2), ((a,q)=1), (1 \leqslant P \leqslant q), then
[
S(a,P;q)=\sum_{1\leqslant x\leqslant P}\exp 2\pi i\,\frac{ax^n}{q}
= Pq^{-1}S(a,q;q)+O\left(q^{1/2+\varepsilon}\right),
\tag{1}
]
where (\varepsilon>0) is arbitrary, and the constant in (O) depends only on (n) and (\varepsilon).
This asymptotic formula for a trigonometric sum; formula (1) is nontrivial if the complete sum (S(a,q;q)) is “large,” i.e. its modulus is greater than (q^{1/2+\varepsilon}); the latter occurs only when (q) contains “many” prime factors to a “high” power. We note that, even in the best case, (1) is an asymptotic formula for (q^{1/2+\varepsilon}\ll P\leqslant q), i.e. if the original sum (S(a,P;q)) is “sufficiently long.” In [3, 4] the author obtained an asymptotic formula for “short” sums (S(a,P;q)), if (q) is a “high” power of a prime number. However, Theorem 1 in [3] and Lemma 4 in [4] were formulated incorrectly. In the present article new asymptotic formulas will be obtained when (q) is a power of a prime number.
Notation and conditions on the parameters used.
(k,l,m,n,t,P,P_1,a,a_1,\ldots,a_n,b) are integers; (n\geqslant 20); (r) is a real number, with (1\leqslant r\leqslant 0.1n); (t\geqslant 4rn); (C,C_1,\ldots,\gamma,\gamma_1,\ldots) are positive absolute constants; (\varepsilon>0) is an arbitrarily small number; (p) is a prime number, (p>\exp cn\ln n), (p/\ln p>t); (q=p^t); (P=q^{1/r}); ((a,p)=1); (\delta_n(x)=1) or (0), according as (x\equiv 0\pmod n) or (x\not\equiv 0\pmod n); the constants in the signs (O) and (\ll) depend only on (n); as above,
[
S(a,P;q)=\sum_{1\leqslant x\leqslant P}\exp 2\pi i\,\frac{ax^n}{q}.
]
Theorem 1. There exist (\varphi(q)(1-2q^{-\varepsilon})) values of (a) such that, for every (P), the asymptotic formula holds
[
S(a,P;q)=Pq^{-1}S(a,q;q)+O\left(\sqrt{Pq^{\varepsilon}\ln^3 q}+P^{3/4}\right).
]
Theorem 2. For every (a) the equality holds:
[
S(a,P;q)=Pq^{-1}S(a,q;q)+O\left(P^{1-\gamma/r^2}\right).
]
Proof of Theorem 1. Let (a) be arbitrary, ((a,q)=1).
1. Separation of the principal term. Define integers (s) and (m) from the conditions (2^s\leqslant P<2^{s+1}), (p^{m-1}\leqslant 2^{0.5s}<p^m). Then (s=\left[\frac{t}{r}\log p\right]), (m-1=(0.5s/\log p)); (m-1<t/2r\leqslant m+1/2t); moreover, (p^m\ll P^{3/4}). Therefore we have
[
S(a,P;q)=
\sum_{1\leqslant x\leqslant P_1p^m}
\exp 2\pi i\,\frac{ax^n}{q}
+O\left(P^{3/4}\right)
=
]
[
\sum_{0\leqslant b\leqslant P_1-1}
\sum_{1\leqslant x\leqslant p^m}
\exp 2\pi i\,\frac{a(x+bp^m)^n}{q}
+O\left(P^{3/4}\right).
\tag{2}
]
Let us now consider the inner sum
[
S^*(a)=\sum_{1\leq x\leq p^m}\exp 2\pi i\,\frac{a(x+bp^m)^n}{q}.
]
Let (t=t_1n+t_2,\ 0\leq t_2\leq n-1,\ t_1=[t/n];\ r_m=t/m); then (r_m\leq 2.001r) and (r_m\geq t/(t/2r+1)\geq 1). We split the sum (S^*(a)) into (t_1-\delta_n(t)+1) sums, grouping together the terms with (x) not divisible by (p), divisible by (p) but not by (p^2), divisible by (p^2) but not by (p^3), etc. From the definition of (r_m) and the conditions on (t) and (r_m) we obtain: (r_m=t/m\leq 0.2001n,\ m\geq 5t/n-1/2t>t_1+1;\ n(t_1-\delta_n(t)+1)\geq t). Consequently,
[
S^(a)=p^{m-t_1+\delta_n(t)-1}+\sum_{0\leq \nu\leq t_1-\delta_n(t)} S_\nu^(a),
\tag{3}
]
where
[
S_\nu^*(a)=
\sum_{\substack{1\leq x\leq p^{m-\nu}\ (x,p)=1}}
\exp 2\pi i\,\frac{a(x+bp^{m-\nu})^n}{p^{t-\nu n}},
\qquad
\nu=0,1,\ldots,t_1-\delta_n(t).
]
Now we split the sum over (\nu) in (3) into two parts:
a) [
0\leq \nu\leq \nu_0=
\left[\frac{t}{n-1}\left(1-\frac{1}{r_m}\right)\right];
]
b) [
\nu_0+1\leq \nu\leq t_1-\delta_n(t).
]
Each of the intervals a) and b) is nonempty. Indeed, case a) is trivial. In case b) we have:
[
\nu_0+1\leq \frac{t}{n-1}\left(1-\frac{1}{r_m}\right)+1
\leq \frac{t}{n}\left(1+\frac{1}{n-1}\times\right)
]
[
\times\left(1-\frac{1}{2.001r}\right)<\frac{t}{n}-1\leq t_1-\delta_n(t).
]
Consider case b). Then (m-\nu\geq t-\nu n); indeed, this follows from the inequality
[
\nu\geq (t-m)/(n-1)=t(1-1/r_m)/(n-1).
]
Consequently, (S_\nu^*(a)) is a complete sum, and
For any (l\geq 2) we have the equality:
[
S_\nu^*(a)=
\sum_{\substack{1\leq x\leq p^{t-\nu n}\ (x,p)=1}}
\exp 2\pi i\,\frac{ax^n}{p^{t-\nu n}}.
]
[
\sum_{\substack{1\leq x\leq p^l\ (x,p)=1}}
\exp \frac{ax^n}{p^l}=0
]
(see, for example, (1), p. 270). Consequently, only (S_{t_1}^*(a)) can be different from zero, and this will occur only when (t_2=1), i.e.
[
S_{t_1}^*(a)=\delta_n(t-1)p^{m-t_1-1}
\sum_{x=1}^{p-1}\exp 2\pi i\,\frac{ax^n}{p}
=
\delta_n(t-1)p^{m-t_1-1}(S(a,p;p)-1).
]
Thus,
[
S^(a)=p^{m-t_1+\delta_n(t)-1}
+\delta_n(t-1)p^{m-t_1-1}(S(a,p;p)-1)+
]
[
+\sum_{0\leq \nu\leq \nu_0}S_\nu^(a)
=p^m q^{-1}S(a,q;q)+\sum_{0\leq \nu\leq \nu_0}S_\nu^*(a),
]
which also follows from (1), p. 270. From the last formula, the definition of (S_\nu^*(a)), and (2) we find
[
S(a,P;q)=Pq^{-1}S(a,q;q)+R(a,P;q)+O(P^{3/4}),
\tag{4}
]
where
[
R(a,P;q)=\sum_{1\leq x\leq P}^{\prime}\exp 2\pi i\,\frac{ax^n}{q},
]
and the prime on the sum means that the summation is over (x\not\equiv 0\pmod{p^{\nu_0+1}}).
2. Estimate of the remainder. The quantity (|R(a,P;q)|) depends essentially on (P). We shall now pass to a larger quantity which will depend only
on (n, a), and (q). Note that (\nu_0) depends only on (n, t), and (s). We have
[
R(a,P;q)=2^{-s-1}\sum_{1\le x\le 2^{s+1}}' \exp 2\pi i\,\frac{ax^n}{q}
\sum_{0\le b\le 2^{s+1}} \sum_{y\le P} \exp 2\pi i\,\frac{b(x-y)}{2^{s+1}},
]
where the prime, as before, means that (x\not\equiv 0 \pmod {p^{\nu_0+1}}). Separating the term with (b=0), summing over (y), and passing to inequalities, we obtain
[
|R(a,P;q)|\ll
\sum_{0\le b\le 2^{s+1}} \frac{1}{b+1}
\left|
\sum_{1\le x\le 2^{s+1}}'
\exp 2\pi i\left(\frac{ax^n}{p^t}-\frac{bx}{2^{s+1}}\right)
\right|.
]
Consequently,
[
|R(a,P;q)|^2\ll \log q
\sum_{0\le b\le 2^{s+1}} \frac{1}{b+1}
\left|
\sum_{1\le x\le 2^{s+1}}'
\exp 2\pi i\left(\frac{ax^n}{p^t}-\frac{bx}{2^{s+1}}\right)
\right|^2,
]
[
P^{-1}|R(a,P;q)|^2\ll
2^{-s-1}\log q
\sum_{0\le b\le 2^{s+1}} \frac{1}{b+1}
\left|
\sum_{1\le x\le 2^{s+1}}'
\exp 2\pi i\left(\frac{ax^n}{p^t}-\frac{bx}{2^{s+1}}\right)
\right|^2
]
[
\ll \log q
\sum_{0\le s\le \log q} 2^{-s-1}
\sum_{0\le b\le 2^{s+1}} \frac{1}{b+1}
\left|
\sum_{1\le x\le 2^{s+1}}'
\exp 2\pi i\left(\frac{ax^n}{p^t}-\frac{bx}{2^{s+1}}\right)
\right|^2
=\log q\cdot \Phi,
]
where (\Phi=\Phi(a,q)=\Phi(n,a,q)\ge 0) and depends only on (n,a), and (q).
Arrange (\Phi(a,q)) in increasing order:
(\Phi(a_1,q)\le \Phi(a_2,q)\le \cdots\le \Phi(a_\varkappa,q)),
(\varkappa=\varphi(q)). For (1\le \omega<\varkappa) we have
[
\Phi(a_\omega,q)\ll
\frac{1}{\varkappa-\omega+1}\sum_{\omega\le m\le \varkappa}\Phi(a_m,q)
\ll
\frac{1}{\varkappa-\omega+1}\sum_{1\le m\le \varkappa}\Phi(a_m,q)\ll
]
[
\ll
\frac{1}{\varkappa-\omega+1}
\sum_{0\le s\le \log q} 2^{-s-1}
\sum_{0\le b\le 2^{s+1}}\frac{1}{b+1}\,qT_s,
]
where (T_s) is the number of solutions of the congruence
(x^n\equiv y^n \pmod {p^t}),
(1\le x,y\le 2^{s+1}),
(x\not\equiv 0\pmod {p^{\nu_0+1}}),
(y\not\equiv 0\pmod {p^{\nu_0+1}}).
Since (2^{s+1}p^{-\nu_0}\ll p^{t-\nu_0 n}), it follows that
[
T_s\ll \sum_{0\le \nu\le \nu_0} 2^{s+1}p^{-\nu}\ll 2^{s+1},
\qquad
\Phi(a_\omega,q)\ll \frac{q\log^2 q}{\varkappa-\omega+1}.
]
If now we take (\omega=\varkappa-q^{1-\varepsilon}), then
[
\Phi(a_\omega,q)\ll q^\varepsilon\log^2 q,\qquad
P^{-1}|R(a_\nu,P;q)|^2\ll q^\varepsilon\log^3 q
]
for (\nu=1,2,\ldots,\varphi(q)-q^{1-\varepsilon}). From this and from (4) we obtain the assertion of Theorem 1.
For the proof of Theorem 2 we shall need the following
Lemma. Let (q_1=p^k), (n\le k\le t), (1\le u\le 0.5n); (P_1^u=q_1),
((a_1,p)=(a_2,p)=\cdots=(a_n,p)=1),
[
S=\sum_{1\le x\le P_1}
\exp 2\pi i\,\frac{a_1x+a_2px^2+\cdots+a_np^{\,n-1}x^n}{q_1}.
]
Then
[
|S|\le c_1P_1^{\,1-\gamma_1/u^2}.
]
This is one of the variants of the theorem of [5], p. 239.
Proof of Theorem 2. Estimate (|R(a,P;q)|) in (4) (see the proof of Theorem 1). We have
[
R(a,P;q)=\sum_{1\le x<P}' \exp 2\pi i\,\frac{ax^n}{q}
=
\sum_{0\le h<P_1-1}\sum_{0\le \nu\le \nu_0}S_\nu^*(a)+O(P^{3/4}).
]
In the sum (S_\nu^*(a)) make the change of summation variable of the form
(x=py+z), (1\le z\le p-1), (0\le y\le p^{m-\nu-1}-1); then
(ax^n=a(py+z)^n=a_0+a_1py+\cdots+a_np^ny^n), where
((a_0,p)=\cdots=(a_n,p)=1) for every (z), (1\le z\le p-1). Further,
[
|S_\nu^*(a)|\le
\sum_{1\le z\le p-1}
\left|
\sum_{0\le y\le p^{m-\nu-1}}
\exp 2\pi i\,\frac{a_1y+a_2py^2+\cdots+a_np^{\,n-1}y^n}{p^{t-\nu n-1}}
\right|.
]
The estimate of the lemma is applicable to the last sum. Let us verify that the conditions of the lemma are satisfied:
(k=t-\nu n-1;\quad k\leq t;\quad k\geq t-\nu_0 n-1\geq t-1-)
[
-\,t\left(1+\frac{1}{n-1}\right)\left(1-\frac{1}{r_m}\right)
\geq
t-1-t\left(1+\frac{1}{n-1}\right)\left(1-\frac{1}{2.001r}\right)\geq n;
]
let (u=(t-\nu n-1)/(m-\nu-1)), then (u\geq 1), since (\nu\leq (t-m)/(n-1)=t(1-1/r_m)/(n-1)); moreover,
(u=r_m(t-\nu n+1)/(t-\nu r_m-r_m)\leq 2r_m), which follows from the inequality
(r_m\leq \frac12(\nu n+t+1)/(\nu+1)=\frac12(n+(t+1-n)/(\nu+1))).
Thus, always (1\leq u\leq 2r_m\leq 4.002\,r<0.5\,n). Applying the estimate of the lemma, we obtain:
[
|S_\nu^*(a)|\ll
p\cdot p^{(m-\nu-1)(1-\gamma_2/u^2)}
=
p^{(m-\nu)(1-\gamma_3/u^2)}
=
p^{(m-\nu)(1-\gamma_4/r_m^2)}.
]
Thus,
[
|R(a,P;q)|\ll
P_1\sum_{0\leq \nu\leq \nu_0}
p^{(m-\nu)(1-\gamma_1/r_m^2)}
+
P^{3/4}
\ll
]
[
\ll
P_1p^{m(1-\gamma_1/r_m^2)}
+
P^{3/4}
\ll
P^{1-\gamma/r^2}.
]
From (4) and the last estimate, the assertion of Theorem 2 follows.
Remark 1. Theorem 1 can be proved for a wider interval of variation of the values of (P), namely (1\leq r\leq 0.5n).
Remark 2. Hua Lo-keng’s theorem and Theorems 1 and 2 establish the asymptotics of the corresponding trigonometric sums for (1\leq r\leq 2), (1\leq r\leq 0.1n), and (1\leq r\leq c\sqrt n), respectively.
Remark 3. Theorems 1 and 2 can be generalized to a broader class of trigonometric sums.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Moscow
Received
11 III 1969
CITED LITERATURE
- I. M. Vinogradov, Selected Works, Publishing House of the Academy of Sciences of the USSR, 1952.
- Hua Lo-keng, On Exponential Sums, Science Record, 1, 1, 1957, p. 1.
- A. A. Karatsuba, DAN, 169, No. 1, 9 (1966).
- A. A. Karatsuba, Tr. Moscow Math. Soc., 18, 77 (1968).
- A. A. Karatsuba, Izv. Academy of Sciences of the USSR, 28, 1, 237 (1964).