A. A. KARATSUBA, N. M. KOROBOV

ON A MEAN-VALUE THEOREM

(Presented by Academician A. N. Kolmogorov, 12 X 1962)

Let \(N_k(P)=N_k^{(P)}(0,\ldots,0)\) and \(N_k^{(P)}(\lambda_1,\ldots,\lambda_n)\) be the number of solutions of the system of equations

\[ x_1^\nu+\cdots+x_k^\nu = y_1^\nu+\cdots+y_k^\nu+\lambda_\nu, \qquad 0\leq x,y\leq P-1 \quad (\nu=1,2,\ldots,n), \tag{1} \]

where \(\lambda_1,\ldots,\lambda_n\) are arbitrary fixed integers.

In the present note a new proof is given of Vinogradov’s mean-value theorem (see \((^{1,2,5})\)), which makes it possible to improve the factor depending on \(n\) in the estimate for the number of solutions of the system (1). This in turn makes it possible to improve somewhat the estimates of Weyl sums obtained in \((^{1,4})\). In its main idea this work is close to \((^3)\).

Lemma. Suppose \(m\geq 1\), \(\mu_1,\ldots,\mu_n\) are integers, \(p>n\) is prime, and \(T\) is the number of solutions of the system of congruences

\[ \left. \begin{aligned} x_1+\cdots+x_n &\equiv \mu_1 \pmod p,\\ &\cdots\cdots\cdots\cdots\\ x_1^n+\cdots+x_n^n &\equiv \mu_n \pmod {p^n}, \end{aligned} \right\} \qquad 0\leq x_j\leq mp^n-1 \]

\[ (j=1,2,\ldots,n), \]

where for \(i\ne j\) the condition \(x_i\not\equiv x_j\pmod p\) is satisfied. Then the estimate

\[ T\leq n!\,m^n p^{n(n-1)/2} \]

is valid.

Proof. Write the quantities \(x_j\) in the form

\[ x_j=x_{j1}+px_{j2}+\cdots+p^{n-1}x_{jn}+p^n x_{j\,n+1} \qquad (j=1,2,\ldots,n), \]

where \(0\leq x_{j\nu}\leq p-1\) \((\nu=1,2,\ldots,n)\) and \(0\leq x_{j\,n+1}\leq m-1\). Note that, according to the additional condition stated in the lemma, for \(i\ne j\) we have \(x_{i1}\ne x_{j1}\). It is easy to see that the quantities \(x_{11},\ldots,x_{n1}\) satisfy the system of congruences

\[ \left. \begin{aligned} x_{11}+\cdots+x_{n1} &\equiv \mu_1\\ &\cdots\cdots\cdots\\ x_{11}^n+\cdots+x_{n1}^n &\equiv \mu_n \end{aligned} \right\} \pmod p. \]

Denote by \(T_1\) the number of solutions of this system. Obviously, \(T_1\leq n!\). Fix one such solution and pass from the original system of congruences to congruences modulo \(p^2\). Then we obtain

\[ \left. \begin{aligned} (x_{11}+px_{12})^2+\cdots+(x_{n1}+px_{n2})^2 &\equiv \mu_2\\ &\cdots\cdots\cdots\cdots\cdots\\ (x_{11}+px_{12})^n+\cdots+(x_{n1}+px_{n2})^n &\equiv \mu_n \end{aligned} \right\} \pmod {p^2}. \]

Hence, after obvious transformations, we obtain a system of \(n-1\) linear congruences in \(x_{12},\ldots,x_{n2}\):

\[ \left. \begin{aligned} x_{11}x_{12}+\cdots+x_{n1}x_{n2} &\equiv \mu_2',\\ x_{11}^{\,n-1}x_{12}+\cdots+x_{n1}^{\,n-1}x_{n2} &\equiv \mu_n' \end{aligned} \right\} \pmod p. \tag{2} \]

Since the quantities \(x_{11},\ldots,x_{n1}\) are distinct, at least \(n-1\) of them are nonzero. For definiteness we shall assume that the quantities \(x_{11},\ldots,x_{n-1,1}\) are nonzero. Then, obviously,

\[ \left| \begin{array}{ccc} x_{11} & \ldots & x_{n-1,1}\\ \cdot & \cdot & \cdot\\ x_{11}^{\,n-1} & \ldots & x_{n-1,1}^{\,n-1} \end{array} \right| = x_{11}\cdots x_{n-1,1} \prod_{1\le i<j\le n-1}(x_{i1}-x_{j1}) \not\equiv 0 \pmod p, \]

and, consequently, for any fixed value of \(x_{n2}\), the quantities \(x_{12},\ldots,x_{n-1,2}\) are determined uniquely from the system (2). Thus, denoting by \(T_2\) the number of solutions of this system, we obtain \(T_2=p\).

Next, considering congruences modulo \(p^3\), we obtain a system of \(n-2\) linear congruences with respect to \(x_{13},\ldots,x_{n3}\); the number of solutions of such a system is \(T_3=p^2\). Similarly we obtain \(T_4=p^3,\ldots,T_n=p^{n-1}\) and \(T_{n+1}=m^n\). Hence, since \(T\le T_1\cdots T_{n+1}\), the assertion of the lemma follows:

\[ T\le n!\,p^{1+2+\cdots+n-1}m^n = n!\,m^n p^{n(n-1)/2}. \]

Now let \(n\ge 2\), \(P\ge (2n)^{2n(1+1/(n-1))^\tau}\), \(0<\theta\le 1\), and \(P_0=P\). Define, for \(\nu=1,2,\ldots,\tau\), the integers \(P_\nu\) and primes \(p_\nu\) by means of the relations

\[ \frac{1}{1+\theta}P_{\nu-1}^{1/n}\le p_\nu\le P_{\nu-1}^{1/n}, \qquad P_\nu=[P_{\nu-1}p_\nu^{-1}]+1. \tag{3} \]

It is easy to show that under these conditions the estimates

\[ n^2<p_\nu,\qquad P_{\nu-1}<p_\nu P_\nu,\qquad P_\nu<(1+2\theta)^n P^{(1-1/n)^\nu}. \tag{4} \]

also hold. We also note that the quantity \(\theta\) is chosen so that in each of the intervals (3) there is a prime number.

Theorem 1. If \(\tau\ge 1\), \(k\ge n^2+n\tau\), then the estimate

\[ N_k(P)<(3k^{2n})^\tau(1+2\theta)^{2k(n+\tau)} P^{2k-n(n+1)/2+n(n+1)/2(1-1/n)^\tau}. \]

holds.

Proof. By definition, \(P=P_0\le p_1P_1\), and, consequently, \(N_k(P)\le N_k(p_1P_1)\), where \(N_k(p_1P_1)\) is the number of solutions of the system of equations (1) with \(\lambda_1=\cdots=\lambda_n=0\):

\[ (x_1+p_1z_1)^\nu+\cdots+(x_k+p_1z_k)^\nu = (y_1+p_1t_1)^\nu+\cdots+(y_k+p_1t_k)^\nu \qquad (\nu=1,2,\ldots,n), \tag{5} \]

\[ 0\le x_j,\ y_j\le p_1-1,\qquad 0\le z_j,\ t_j\le P_1-1. \]

We shall assign the system \(x_1,\ldots,x_k\) to the first class if in it one can find \(n\) distinct quantities \(x_j\). All the remaining systems \(x_1,\ldots,x_k\) will be assigned to the second class. Obviously,

\[ N_k(p_1P_1)=N_k^{(1)}+N_k^{(2)}, \]

where \(N_k^{(1)}\) is the number of solutions of the system (5) for which \(x_1,\ldots,x_k\) and \(y_1,\ldots,y_k\) belong to the first class, while \(N_k^{(2)}\) is the number of solutions of the system (5) for which \(x_1,\ldots,x_k\) or \(y_1,\ldots,y_k\) belong to the second class.

Introduce, for systems with distinct \(x_1,\ldots,x_n\) and arbitrary \(x_{n+1},\ldots,x_k\), the notation \((x_1,\ldots,x_n)x_{n+1},\ldots,x_k\). We shall call permutations of these systems such systems in which the distinct \(x_j\) stand no longer in the first \(n\) places, but in arbitrary places.

It is easy to see that every system of the first class occurs among the permutations of systems of the form \((x_1,\ldots,x_n)x_{n+1},\ldots,x_k\). Consequently, the quantity \(N_k^{(1)}\) does not exceed the number of solutions of the system (5) with variables of the form

\[ (x_1,\ldots,x_n)x_{n+1},\ldots,x_k,\quad (y_1,\ldots,y_n)y_{n+1},\ldots,y_k, \]

multiplied by \((C_k^n)^2\).

But then, using the notation

\[ S(x)=\sum_{z=0}^{p_1-1} e^{2\pi i f(x+p_1z)},\qquad f(x)=\alpha_1 x+\cdots+\alpha_n x^n, \]

we obtain

\[ \begin{aligned} N_k^{(1)} &\leq (C_k^n)^2 \int_0^1\cdots\int_0^1 \left| \sum_{\substack{(x_1,\ldots,x_n)\\ x_{n+1},\ldots,x_k}} S(x_1)\cdots S(x_k) \right|^2\,d\alpha_1\cdots d\alpha_n \leq \\ &\leq (C_k^n)^2 p_1^{2k-2n-1} \int_0^1\cdots\int_0^1 \left| \sum_{(x_1,\ldots,x_n)} S(x_1)\cdots S(x_n) \right|^2 \sum_{x=0}^{p_1-1}|S(x)|^{2k-2n}\,d\alpha_1\cdots d\alpha_n . \end{aligned} \]

The integral appearing on the right-hand side of this inequality is, obviously, equal to the number of solutions of the system

\[ (x_1+p_1z_1)^\nu+\cdots-(y_n+p_1t_n)^\nu =(x+p_1z_{n+1})^\nu+\cdots-(x+p_1t_k)^\nu \]

\[ (\nu=1,2,\ldots,n), \]

or, equivalently, to the number of solutions of the system

\[ (x_1-x+p_1z_1)^\nu+\cdots-(y_n-x+p_1t_n)^\nu =p_1^\nu(z_{n+1}^\nu+\cdots-t_k^\nu) \tag{6} \]

\[ (\nu=1,2,\ldots,n), \]

where \(0\leq x,x_j,y_j\leq p_1-1\), \(0\leq z_j,t_j\leq P_1-1\), and for \(i\ne j\) the condition \(x_i\ne x_j\) is satisfied. Under the same conditions for the range of variation of the unknowns, let us also consider the system of equations

\[ (x_1-x+p_1z_1)^\nu+\cdots-(y_n-x+p_1t_n)^\nu =\lambda_\nu p_1^\nu\qquad (\nu=1,2,\ldots,n), \tag{7} \]

where \(\lambda_1,\ldots,\lambda_n\) are arbitrary fixed integers, and the system of congruences

\[ (x_1-x+p_1z_1)^\nu+\cdots-(y_n-x+p_1t_n)^\nu \equiv 0 \pmod{p_1^\nu}\qquad (\nu=1,2,\ldots,n). \tag{8} \]

Denoting respectively by \(N'_k\), \(N'_n(\lambda_1p_1,\ldots,\lambda_np_1^n)\), and \(T'\) the numbers of solutions of the systems (6), (7), and (8), we obtain

\[ N'_k= \sum_{\lambda_1,\ldots,\lambda_n} N'_n(\lambda_1p_1,\ldots,\lambda_np_1^n)\, N_{k-n}^{(P_1)}(\lambda_1,\ldots,\lambda_n)\leq \]

\[ \leq N_{k-n}(P_1) \sum_{\lambda_1,\ldots,\lambda_n} N'_n(\lambda_1p_1,\ldots,\lambda_np_1^n) = T'N_{k-n}(P_1)* . \]

Finally, choosing in the lemma \(m=[P_1p_1^{-n+1}]+1\), we obtain

\[ T'\leq p_1(p_1P_1)^n n!\bigl([P_1p_1^{-n+1}]+1\bigr)^n p_1^{n(n-1)/2} \leq n!2^nP_1^{2n}p_1^{-n(n+1)/2+2n+1}, \]

\[ N_k^{(1)}\leq (C_k^n)^2p_1^{2k-2n-1}N'_k \leq 2k^{2n}P_1^{2n}p_1^{2k-n(n+1)/2}N_{k-n}(P_1). \tag{9} \]

Let us now estimate the quantity \(N_k^{(2)}\). Obviously,

\[ N_k^{(2)} = \int_0^1\cdots\int_0^1 \left[ \sum_{x_1,\ldots,y_k} S(x_1)\cdots S(y_k) \right]\,d\alpha_1\cdots d\alpha_n, \]

\[ \text{* The sum } \sum_{\lambda_1,\ldots,\lambda_n} \text{ is extended over the region } |\lambda_1|\leq kP_1,\ldots,|\lambda_n|\leq kP_1^n . \]

where the quantities \(x_1,\ldots,y_k\) vary in such a way that at least one of the systems \(x_1,\ldots,x_k;\ y_1,\ldots,y_k\) belongs to the second class. Noting that the number of systems of the second class does not exceed \(n^k p_1^{\,n-1}\), we obtain

\[ \sum_{x_1,\ldots,y_k} S(x_1)\cdots \overline{S(y_k)} \ll n^k p_1^{\,k+n-1}\sum_{x=0}^{p_1-1}|S(x)|^{2k} \ll n^k P_1^{2n}p_1^{\,k+n-1}\sum_{x=0}^{p_1-1}|S(x)|^{2k-2n}, \]

\[ N_k^{(2)} \ll n^k P_1^{2n}p_1^{\,k+n-1} \int_0^1\cdots\int_0^1 \sum_{x=0}^{p_1-1}|S(x)|^{2k-2n}\,d\alpha_1\cdots d\alpha_n = n^k P_1^{2n}p_1^{\,k+n}N_{k-n}(P_1). \]

Since \(n<p_1^{1/2}\) and \(k\ge n^2+n\), we have
\(n^k p_1^{\,k+n}\ll n^{2n}p_1^{\,2k-n(n+1)/2}\), and consequently

\[ N_k^{(2)}\ll n^{2n}P_1^{2n}p_1^{\,2k-n(n+1)/2}N_{k-n}(P_1). \]

Hence, by virtue of (9), it follows that

\[ N_k(P)\ll N_k^{(1)}+N_k^{(2)} \ll 3k^{2n}P_1^{2n}p_1^{\,2k-n(n+1)/2}N_{k-n}(P_1). \]

Applying this inequality \(\tau\) times, we obtain

\[ N_k(P)\ll (3k^{2n})^\tau (P_1\cdots P_\tau)^{2n} (p_1\cdots p_\tau)^{2k-n(n+1)/2} (p_2p_3^2\cdots p_\tau^{\tau-1})^{-2n} N_{k-n\tau}(P_\tau). \tag{10} \]

Using the estimates (4), it is not hard to verify that

\[ P_1\cdots P_\tau\ll p_2p_3^2\cdots p_\tau^{\tau-1}P_\tau^\tau, \qquad p_1\cdots p_\tau\ll (1+2\theta)^\tau P^{1-(1-1/n)^\tau}. \]

Substituting these estimates into (10) and using the trivial estimate for \(N_{k-n\tau}(P_\tau)\), we obtain the assertion of the theorem:

\[ N_k(P)\ll (3k^{2n})^\tau P_\tau^{2k}(p_1\cdots p_\tau)^{2k-n(n+1)/2}\ll \]

\[ \ll (3k^{2n})^\tau(1+2\theta)^{2k(n+\tau)} P^{2k-n(n+1)/2+n(n+1)/2(1-1/n)^\tau}. \]

It follows from Bertrand’s postulate that in Theorem 1 one may choose \(\theta=1\). Using deeper facts from elementary number theory, one can arrange that the quantity \(\theta\) decreases as \(n\) increases. In particular, choosing, for \(n\ge 11\), \(\theta=1/3\ln 2n\) and estimating \(N_{k-n\tau}(P_\tau)\) more precisely, we obtain the following assertion:

Theorem 2. There exist absolute constants \(\alpha\) and \(C\) such that, for \(n\ge 11\), \(1\le \tau\le 2n\ln(n+1)+1\), \(k\ge 2n^2+n\tau\), \(P\ge n^{\alpha n(1+1/(n-1))^\tau}\), the estimate

\[ N_k(P)\le C P^{2k-n(n+1)/2+n(n+1)/2(1-1/n)^\tau}{}^* \]

is valid.

Received
12 X 1962

References

\(^{1}\) I. M. Vinogradov, Izv. Acad. Sci. USSR, Ser. Math., 15, 109 (1951).
\(^{2}\) I. M. Vinogradov, Izv. Acad. Sci. USSR, Ser. Math., 22, 161 (1958).
\(^{3}\) A. A. Karatsuba, Vestn. Moscow Univ., 4, 28 (1962).
\(^{4}\) N. M. Korobov, DAN, 123, No. 1, 28 (1958).
\(^{5}\) Hua Loo-keng, Quart. J. Math., 20, 48 (1949).

\[ \text{* It can be shown that for } \alpha=86 \text{ one obtains } C\le 1. \]