MATHEMATICS

N. M. KOROBOV

PROPERTIES AND COMPUTATION OF OPTIMAL COEFFICIENTS

(Presented by Academician I. M. Vinogradov, 5 II 1960)

We shall say that the function

[
f(x_1,\ldots,x_s)=\sum_{m_1,\ldots,m_s=-\infty}^{\infty}
C(m_1,\ldots,m_s)e^{2\pi i(m_1x_1+\ldots+m_sx_s)}
\tag{1}
]

belongs to the class (E_s^\alpha), if
(C(m_1,\ldots,m_s)=O((\overline m_1\cdots \overline m_s)^{-\alpha})), where
(\alpha>1) and (\overline m_\nu=\max(1,|m_\nu|)). Denote by (-R) the error of the quadrature formula

[
\int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\cdots dx_s
=
\frac1p\sum_{k=1}^p f\left(\frac{ka_1}{p},\ldots,\frac{ka_s}{p}\right)-R.
\tag{2}
]

According to ((^1,^2)), the integers (a_\nu=a_\nu(p)) ((\nu=1,2,\ldots,s)) are called optimal coefficients if, for functions (f\in E_s^\alpha), the estimate
(R=O(p^\beta\ln^\beta p)) holds, where (\beta) and the constant in (O) do not depend on (p).

In paper ((^1)) the existence was proved and a method was given for finding optimal coefficients for which (\beta=\alpha s). In the present paper other methods for computing optimal coefficients are indicated, and some of their properties are presented, with applications both in approximate analysis and directly in number theory.

Let (z) be an integer, (p>s) a prime, ([A]) the integral part and ({A}) the fractional part of the number (A). Define the function (H(z)) by the equality

[
H(z)=\frac{3^s}{p}\sum_{k=1}^p
\left(1-2\left{\frac{k}{p}\right}\right)^2\cdots
\left(1-2\left{\frac{kz^{s-1}}{p}\right}\right)^2.
\tag{3}
]

Theorem 1. If the minimum of the function (H(z)) on the interval (1\le z<p) is attained at (z=a), then the integers (1,a,\ldots,a^{s-1}) will be optimal coefficients for those classes (E_s^\alpha) for which (\alpha\ge2).

Proof. Denote by (\delta_p(m)) one or zero according as the congruence (m\equiv0\pmod p) holds or not. We use the equalities

[
\delta_p(m)=\frac1p\sum_{k=1}^p e^{\frac{2\pi i mk}{p}},
\qquad
3(1-2{x})^2=\sum_{m=-\infty}^{\infty}\frac{e^{2\pi i mx}}{\psi(m)},
\tag{4}
]

where, for (m\ne0), (\psi(m)=\frac{\pi^2}{6}m^2), and (\psi(0)=1). From (3), after transformations analogous to those carried out in ((^1)), since (\psi(m)\ge \overline m^2), we obtain

[
H(z)-1=
\sum_{-\infty}^{\infty}{}'
\frac{\delta_p(m_1+\cdots+m_s z^{s-1})}{\psi(m_1)\cdots\psi(m_s)}
\le
\sum_{-(p-1)}^{p-1}{}'
\frac{\delta_p(m_1+\cdots+m_s z^{s-1})}{(\overline m_1\cdots \overline m_s)^2}
+O\left(\frac1{p^2}\right),
]

where (\sum') means summation over systems ((m_1,\ldots,m_s)\ne(0,\ldots,0)), in which respectively (|m_\nu|<\infty) or (|m_\nu|\le p-1) ((\nu=1,2,\ldots,s)).

Let (d) be the greatest common divisor of the numbers (m_1,\ldots,m_s), and let (A(m_1,\ldots,m_s)) be the number of solutions of the congruence
(m_1+\cdots+m_s z^{s-1}\equiv0\pmod p). Thus

since for (d \not\equiv 0 \pmod p) the estimate (A(m_1,\ldots,m_s)\leqslant s-1) is valid, we have

[
\min_{1\leqslant z<p}\sum_{-(p-1)}^{p-1}{}'
\frac{\delta_p(m_1+\cdots+m_s z^{s-1})}{\bar m_1\cdots \bar m_s}
\leqslant
\frac{1}{p-1}\sum_{-(p-1)}^{p-1}{}'
\frac{1}{\bar m_1\cdots \bar m_s}
\sum_{z=1}^{p-1}\delta_p(m_1+\cdots
]

[
\cdots+m_s z^{s-1})
\leqslant
\frac{1}{p-1}\sum_{-(p-1)}^{p-1}{}'
\frac{A(m_1,\ldots,m_s)}{\bar m_1\cdots \bar m_s}
=
O!\left(\frac{\ln^s p}{p}\right)
]

and, consequently,

[
H(a)-1\leqslant
\left[
\min_{1\leqslant z<p}\sum_{-(p-1)}^{p-1}{}'
\frac{\delta_p(m_1+\cdots+m_s z^{s-1})}{\bar m_1\cdots \bar m_s}
\right]^2
+O!\left(\frac{1}{p^2}\right)
=
O!\left(\frac{\ln^{2s}p}{p^2}\right).
\tag{5}
]

For (\alpha\geqslant 2), from (1) and (2), applying the first of the equalities (4), we obtain

[
R=
\sum_{m_1,\ldots,m_s=-\infty}^{\infty}{}'
C(m_1,\ldots,m_s)\delta_p(a_1m_1+\cdots+a_sm_s),
\tag{6}
]

[
|R|\leqslant C\sum_{-\infty}^{\infty}{}'
\frac{\delta_p(a_1m_1+\cdots+a_sm_s)}{(\bar m_1\cdots \bar m_s)^\alpha}
\leqslant
C\left[
\sum_{-\infty}^{\infty}{}'
\frac{\delta_p(a_1m_1+\cdots+a_sm_s)}{(\bar m_1\cdots \bar m_s)^2}
\right]^{\alpha/2},
]

where (C=C(\alpha,s)). Hence, since (\bar m^2\geqslant \dfrac{6}{\pi^2}\psi(m)), for (a_\nu=a^{\nu-1}), by virtue of (5) the assertion of the theorem follows:

[
\sum_{-\infty}^{\infty}{}'
\frac{\delta_p(m_1+\cdots+m_sa^{s-1})}{(\bar m_1\cdots \bar m_s)^2}
\leqslant
\left(\frac{\pi^2}{6}\right)^s
\sum_{-\infty}^{\infty}{}'
\frac{\delta_p(m_1+\cdots+m_sa^{s-1})}{\psi(m_1)\cdots\psi(m_s)}
=
\left(\frac{\pi^2}{6}\right)^s[H(a)-1],
\tag{7}
]

[
|R|\leqslant
C\left(\frac{\pi^2}{6}\right)^{\alpha s/2}
[H(a)-1]^{\alpha/2}
=
O!\left(\frac{\ln^{\alpha s}p}{p^\alpha}\right).
]

Corollary. For (s\geqslant 2), the incomplete quotients of the expansions of the numbers

[
\left{\frac{a^2}{p}\right},\ldots,\left{\frac{a^{s-1}}{p}\right}
]

into continued fractions are bounded by (C\ln^s p), where the constant (C) depends only on (s).

Indeed, since for (|m_1|\leqslant p/2), from the congruence (-m_1\equiv a^\nu m_{\nu+1}\pmod p) there follows the equality
[
|m_1|=p\left(\frac{a^\nu m_{\nu+1}}{p}\right),
]
where ((A)) is the distance from (A) to the nearest integer, by virtue of (7) and (5) we obtain

[
\frac{1}{p^2}
\sum_{|m_{\nu+1}|\ne 0\;(\mathrm{mod}\;p)}
\left(
\left(\frac{a^\nu m_{\nu+1}}{p}\right)m_{\nu+1}
\right)^{-2}
=
\sum_{1\leqslant |m_1|<p/2}
\sum_{|m_{\nu+1}|=1}^{\infty}
\frac{\delta_p(m_1+a^\nu m_{\nu+1})}{(|m_1|\,|m_{\nu+1}|)^2}
\leqslant
]

[
\leqslant
\sum_{-\infty}^{\infty}{}'
\frac{\delta_p(m_1+\cdots+m_sa^{s-1})}{(\bar m_1\cdots \bar m_s)^2}
=
O!\left(\frac{\ln^{2s}p}{p^2}\right).
]

Hence, for any integer (m_{\nu+1}) not divisible by (p), for some (C=C(s)) we obtain the inequalities

[
\left(\frac{a^\nu}{p}m_{\nu+1}\right)
\geqslant
\frac{1}{C|m_{\nu+1}|\ln^s p}
\qquad
(\nu=1,2,\ldots,s-1),
\tag{8}
]

which are equivalent to the assertion of the corollary.

The following theorem somewhat strengthens this result.

Theorem 2. For (s\geqslant 2), for each sufficiently large prime (p) one can indicate no fewer than (2^{-s+1}p) integers (a=a(p)) such that the in-

the incomplete quotients (\dfrac{a}{p}, \left{\dfrac{a^2}{p}\right}, \ldots, \left{\dfrac{a^{s-1}}{p}\right}) are bounded respectively by the quantities (2(5\ln p), 2(5\ln p)^2, \ldots, 2(5\ln p)^{s-1}).

Proof. For (\nu=1,2,\ldots,s-1) introduce the notation

[
p_\nu=\left[\frac{p}{2^\nu}\right],\qquad
q_\nu=\left[\frac{p}{2\cdot 5^\nu \ln^\nu p}\right]+1,\qquad
S_\nu(z)=\sum_{\overline{m_1}\ldots \overline{m_{\nu+1}}<q_\nu}^{\prime}
\delta_p(m_1+\cdots+m_{\nu+1}z^\nu).
]

Let (z_1,\ldots,z_{p-1}) be such a permutation of the numbers (1,2,\ldots,p-1) for which
[
S_1(z_1)\geq S_1(z_2)\geq\cdots\geq S_1(z_{p-1}).
]
Then, for sufficiently large (p),

[
S_1(z_{p_1})\leq \frac1{p_1}\sum_{j=1}^{p_1}S_1(z_j)\leq
\frac1{p_1}\sum_{\overline{m_1m_2}<q_1}^{\prime}1<1,
]

and, by the definition of (S_\nu(z)), we obtain
[
S_1(z_{p_1})=\cdots=S_1(z_{p-1})=0.
]
It follows that for (a=z_j), where (p_1\leq j<p), for nontrivial solutions of the congruence
[
m_1+am_2\equiv 0 \pmod p
]
the inequality (\overline{m_1m_2}\geq q_1) is satisfied. But then, since
[
|m_1|=\left(\frac{am_2}{p}\right)p,
]
for (m_2\not\equiv 0\pmod p) we obtain

[
\left(\frac{am_2}{p}\right)\geq \frac{q_1}{p|m_2|}>
\frac{1}{10|m_2|\ln p}.
]

From this inequality, since (p-p_1\geq p_1), we obtain the assertion of the theorem for the case (s=2).

We apply induction. Suppose the theorem is true for some (s=k) ((k\geq2)). Then on the interval ([1,p-1]) there exist (p_{k-1}) values of (a) for which, for (\nu=1,2,\ldots,k-1), the incomplete quotients (\left{\dfrac{a^\nu}{p}\right}) are bounded respectively by the quantities (2\cdot 5^\nu \ln^\nu p). Arranging these values (a) so that the inequalities
[
S_k(z'1)\geq\cdots\geq S_k(z')}
]
hold, as above, we obtain
[
S_k(z'{p_k})=\cdots=S_k(z')=0.}
]
Choose (a=z'j), where (p_k\leq j\leq p). Then for nontrivial solutions of the congruence
[
m_1+\cdots+m_{k+1}a^k\equiv 0 \pmod p
]
the inequality (\overline{m_1\cdots m_{k+1}}\geq q_k) will be satisfied. In particular, for solutions in which
[
m_2+\cdots+m_{k+1}a^{k-1}\not\equiv 0 \pmod p,
]
for (p>p_0) we obtain

[
\left(\frac{a}{p}m_2+\cdots+\frac{a^k}{p}m_{k+1}\right)\geq
\frac{q_k}{p\overline{m_2}\cdots \overline{m_{k+1}}}>
\frac{1}{2\cdot 5^k\,\overline{m_2}\cdots \overline{m_{k+1}}\ln^k p}.
\tag{9}
]

Hence, putting (m_2=\cdots=m_k=0), it follows that the incomplete quotients
[
\left{\frac{a^k}{p}\right}
]
are bounded by the quantity (2\cdot 5^k\ln^k p), and, since (p_{k-1}-p_k+1>p_k), we obtain the assertion of the theorem.

Inequalities (8) and (9) may be regarded as consequences of a more general assertion connecting the concept of optimal coefficients with questions of linear Diophantine approximations.

Theorem 3. Let ((a_\nu,p)=1) and (a_\nu b_\nu\equiv 1\pmod p) ((\nu=1,2,\ldots,s)). A necessary and sufficient condition for the quantities (a_1,\ldots,a_s) to be optimal coefficients is the fulfillment of the inequalities

[
\left|
\frac{b_{\nu-1}a_\nu}{p}m_\nu+\cdots+
\frac{b_{\nu-1}a_s}{p}m_s-n
\right|>
\frac{B}{\overline{m_\nu}\cdots \overline{m_s}^{\,1}\ln^\gamma p}
\qquad (\nu=2,3,\ldots,s)
\tag{10}
]

for any integers (n,m_\nu,\ldots,m_s) satisfying the condition
[
a_\nu m_\nu+\cdots+a_s m_s\not\equiv0\pmod p,
]
where (B=B(s)>0) and (\gamma=\gamma(s)\geq0).

The proof is based on the use of the relation

[
R=\sum_{-\infty}^{\infty}
\frac{\delta_p(a_1m_1+\cdots+a_sm_s)}
{(\overline{m_1}\cdots \overline{m_s})^\alpha},
]

which is obtained from (6) with
[
C(m_1,\ldots,m_s)=(\overline{m_1}\cdots \overline{m_s})^{-\alpha};
]
in proving the sufficiency of conditions (10), Theorem 2 of the paper ((^3)) is also used.

Corollary. Let ((a,p)=1), (s \geqslant 2), (B=B(s)>0), (\gamma=\gamma(s)\geqslant 0), and let (m_2,\ldots,m_s) be arbitrary integers for which (m_2+\cdots+a^{s-2}m_s \not\equiv 0\pmod p). The integers (1,a,\ldots,a^{s-1}) will then and only then be optimal coefficients when the condition

[
\left(\frac{a}{p}m_2+\cdots+\frac{a^{s-1}}{p}m_s\right)>
-\frac{B}{\overline{m}_2\ldots \overline{m}_s\ln^\gamma p}
\tag{11}
]

is satisfied.

Remark 1. Using Theorem 3, it is easy to show that the optimality property of the coefficients indicated in Theorem 1 for (\alpha\geqslant 2) extends to all classes (E_s^\alpha) with (\alpha>1).

Remark 2. The process of finding integers (a=a(p)) leading to inequality (9) can be used in computing optimal coefficients. To simplify the computations, we modify this process as follows. Let

[
\sigma_\nu(z)=
\sum_{\overline{m}_1\ldots \overline{m}_s=\nu}
\delta_p(m_1+\cdots+m_s z^{s-1}),
]

where (p>s) is prime. Those of the integers (z\in[1,p-1]) for which (\sigma_1(z)=0) we denote by (z_1). Next, those among the (z_1) for which (\sigma_2(z_1)=0) we denote by (z_2). In general, those among the (z_{\nu-1}) for which (\sigma_\nu(z_{\nu-1})=0) we denote by (z_\nu). Obviously, (\sigma_1(z_\nu)=\cdots=\sigma_\nu(z_\nu)=0). The process terminates after obtaining quantities (z_n), for each of which (\sigma_{n+1}(z_n)>0). It is easy to verify that if (a) is equal to any of the quantities (z_n), then, by virtue of (11), the integers (1,a,\ldots,a^{s-1}) will be optimal coefficients.

The number of operations necessary for computing optimal coefficients can be significantly reduced by passing to quantities (p=p'p''), where (p') and (p'') are prime and (p'') has order (\sqrt{p'}). Let, for example,

[
\widetilde{H}(z)=
\frac{3^s}{p'p''}\sum_{k=1}^{p'p''}
\left(1-2\left{\frac{p'+p''}{p'p''}k\right}\right)^2
\cdots
\left(1-2\left{\frac{p'z^{s-1}+p''a^{s-1}}{p'p''}k\right}\right)^2,
]

where (a) is defined as in Theorem 1 for (p=p'). Let, further, (\widetilde{H}(b)=\min_{1\leqslant z