A. F. LAVRIK

ON THE PROBLEM OF DIVISORS IN SEGMENTS OF ARITHMETIC PROGRESSIONS

(Presented by Academician I. M. Vinogradov on April 5, 1965)

§ 1. The problem of divisors in segments of arithmetic progressions, which has numerous and important applications, consists in deriving an asymptotic formula for sums of the form

\[ \sum_{dn+l\le x}\tau_k(dn+l), \tag{1} \]

with \(d\le x^\gamma,\ 0\le l\le d\) for as large \(\gamma\) as possible, where \(\tau_k(m)\) denotes the number of all decompositions of \(m\) into \(k\) natural factors.

For \(k=4,\ d\le \sqrt[5]{x}\), with \(\sqrt[5]{x}\) close to \(\sqrt{x}\), this problem was solved by Yu. V. Linnik in \((^{1})\) on the basis of the method of the shortened functional equation for Dirichlet \(L\)-functions.

In the present paper results are given which generalize and further develop the ideas of this work of Yu. V. Linnik.

For sums (1) the following holds.

Theorem 1. For integers \(k\ge 4,\ 4\le 2m\le k,\ (l,D)=1\), uniformly in \(D>1,\ 0\le l<D\),

\[ \sum_{\substack{n\le x\\ n\equiv l\pmod D}}\tau_k(n) = \frac{x}{\varphi(D)} \sum_{d\mid D^k} \sum_{d=d_1\ldots d_k} \frac{\mu(d_1)\ldots\mu(d_k)}{d} P_k\!\left(\ln\frac{x}{d}\right) +R, \tag{2} \]

where

\[ R=R_{k,m}(x,D)\ll \frac{1}{\varphi(D)} x^{1-1/2^\nu m} D^{(3k+2m)/2^\nu+3m+\varepsilon_0} \ln^b x; \]

\(\varphi\) is Euler’s function; \(\mu\) is the Möbius function; \(P_k\) is a polynomial of degree \(k-1\), determined by the equality

\[ P_k\!\left(\ln\frac{x}{d}\right) = \frac{d}{(k-1)!\,x} \lim_{s\to 1} \frac{d^{k-1}}{ds^{k-1}} \left\{ \frac{\zeta^k(s)(s-1)^k}{s} \left(\frac{x}{d}\right)^s \right\}; \]

\(\zeta\) is the Riemann zeta-function; \(\nu\) is an integer \(\ge (2k-3m)/m\); \(\varepsilon=0\) or an arbitrarily small \(\varepsilon>0\), according as \(2m=k\) or \(2m<k\); \(b\) is a constant depending only on \(k\).

Since the main term in (2) is of order \(x\ln^{k-1}x/\varphi(D)\), (2) gives for the asymptotics of sums (1) the range

\[ D\ll \frac{x^\gamma}{\ln^b x}, \qquad \gamma=\frac{8}{3k+2m}-\varepsilon_0, \]

and, correspondingly, a reduction in the remainder term by

\[ x^{-\beta}, \qquad \beta= \frac{1}{2^\nu m} \left[ 1-\gamma\left(\frac{3k+2m}{8}+\varepsilon\right) \right]. \]

It is seen from this that the best range for \(D\) is attained when \(m=2\), with \(\gamma=8/(3k+4)-\varepsilon_0\), and that in this case the reduction of the remainder with increasing \(k\) decre—

decreases according to a power law \((\nu=k-3)\). In the other extreme case \(2m=k\) we have a bound for \(D\) with \(\gamma=2/k\), but the reduction of the remainder with increasing \(k\) decreases only as \(x^{-\beta}\), with \(\beta=1/k-\gamma/2\).

Let us note that the case of primitive progressions, considered in Theorem 1, is the basic one, since any other case is reduced to it in an entirely elementary way \((^{1,2})\).

§ 2. By means of a new truncated equation, Yu. V. Linnik \((^1)\) achieved a fundamental advance in estimates of \(L(1/2+it,\chi)\) on the real axis and near it \((|t|<\mathrm{const})\).

A further development of the idea of this equation is the following

Theorem 2. If \(\chi\) is a primitive character, \(d>1\), \(z\ne0\), \(|\arg z|\le \pi/2\), then in the whole plane of the variable \(s\), except for the points \(s=0,-1,-2,\ldots\), the equation holds
\[ \Gamma\left(\frac{s+a}{2}\right)L(s,\chi) = \sum_{n=1}^{\infty}\frac{\chi(n)}{n^s} \Gamma\left(\frac{s+a}{2},\frac{\pi n^2 z}{d}\right) + \]
\[ + \varepsilon(\chi)\left(\frac{d}{\pi}\right)^{1/2-s} \sum_{n=1}^{\infty}\frac{\overline{\chi}(n)}{n^{1-s}} \Gamma\left(\frac{1-s+a}{2},\frac{\pi n^2}{dz}\right), \tag{3} \]
where \(\varepsilon(\chi)\) and \(a\) denote the same as in the usual equation for \(L(s,\chi)\);
\[ \Gamma(a,u)=\int_u^\infty e^{-x}x^{a-1}\,dx;\qquad \Gamma(a)\text{ is the gamma function.} \]

The meaning of the new form (3) of the functional equation for \(L(s,\chi)\) is that this equation is, in essence, truncated and makes it possible to obtain estimates of these functions in the critical strip, uniformly in \(d\) and \(s\). Namely, if one takes
\[ z=\exp i\,\operatorname{sign}\operatorname{Im}s\cdot \left(\frac{\pi}{2}-\frac{1}{|\operatorname{Im}s|+1}\right), \]
then the left-hand side of (3) will be determined (up to an accuracy \(O(1)\)) by segments of the series on the right-hand side with \(n\le N\), where
\[ N=\sqrt{d(|t|+1)\ln d(|t|+1)}, \]
and in this process from each term there will be separated a factor canceling the growth of the function \(\Gamma^{-1}((s+a)/2)\).

More precisely, the following is true.

Theorem 3. If \(\chi\) is a primitive character modulo \(d>1\), \(s=\sigma+it\), \(0<\sigma<1\), \(t\) real, then
\[ L(s,\chi)= \sum_{n\le N}\frac{\chi(n)}{n^s}c_n(s,d,a) + \]
\[ + \varepsilon(\chi)\left(\frac{d}{\pi}\right)^{1/2-s} \frac{\Gamma((1-s+a)/2)}{\Gamma((s+a)/2)} \sum_{n\le N}\frac{\overline{\chi}(n)}{n^{1-s}}c_n(1-s,d,a) + O(N^{-b}), \]
where \(c_n(\omega,d,a)\) is a quantity bounded in all its arguments by an absolute constant; \(b\) is an arbitrary positive constant.

From Theorem 2 there also follows

Theorem 4. Let \(m\) be an integer,
\[ N_m=\sqrt{\frac{d|t|}{2\pi}}+m\sqrt{\frac{d}{2\pi}}, \qquad N=d(|t|+1), \qquad T=\sqrt{|t|+1}\ln N. \]
Then, for a primitive character \(\chi\) modulo \(d>1\), \(s=\sigma+it\), \(0<\sigma<1\), and arbitrary real \(t\),
\[ L(s,\chi)= A\sum_{|m|\le T} \left\{ \max_{v\le N_m} \left| \sum_{\substack{N_{m-1}<n\le v\\ n>0}} \frac{\chi(n)}{n^s} \right| + \right. \]
\[ \left. + BN^{1/2-\sigma}\max_{u\le N_m} \left| \sum_{\substack{N_{m-1}<n\le u\\ n>0}} \frac{\overline{\chi}(n)}{n^{1-s}} \right| \right\} + O(N^{-b}), \]

where \(A, B\) are quantities bounded by absolute constants, and \(b\) is an arbitrary positive constant.

The latter equation has an especially simple form when \(\sigma=1/2\). Namely, if \(s=1/2+it\), then

\[ L(s,\chi)=A \sum_{|m|\le T}\max_{v\le Nm} \left| \sum_{\substack{N_{m-1}<n\le v\\ n>0}} \frac{\chi(n)}{n^s} \right| +O(N^{-b}). \]

Thus, instead of a complete sum over \(1\le n\le \sqrt d\,|t|\ln d|t|\), we have \(T=\sqrt{|t|}\ln d|t|\) separate, nonoverlapping intervals of length \(<\sqrt d\).

The derivation of Theorem 2 rests on the usual functional equation for \(L(s,\chi)\). Let us note that, in view of the symmetry of the right-hand side of (3), this equation, in turn, follows in an obvious way from (3). As special cases, (3) contains the equation of P. O. Kuz’min \({}^{3}\) (\(z=i\)) and its generalization from \({}^{4}\) (\(z=\delta i\)), as well as the equations indicated in \({}^{1,2,5}\); moreover, the case \(z=\delta i\), \(\delta>0\), is apparently not the best one, because for such values of \(z\) the series on the right-hand side of (3) cease to be absolutely convergent.

The derivation of Theorem 1 is carried out on the basis of the approximate equation of Theorem 3 with the aid of D. A. Burgess’s estimate \({}^{6}\) for character sums. Less precise results were indicated earlier \({}^{2,7}\)*.

Received
4 XI 1964

CITED LITERATURE

\({}^{1}\) Yu. V. Linnik, Matem. sborn., 53 (95), No. 1, 3 (1961).
\({}^{2}\) A. F. Lavrik, DAN, 154, No. 1, 34 (1964).
\({}^{3}\) P. O. Kuz’min, Izv. AN SSSR, ser. matem., 10, 1472 (1934).
\({}^{4}\) Yu. V. Linnik, Izv. AN SSSR, ser. matem., 24, No. 5, 629 (1960).
\({}^{5}\) A. F. Lavrik, Izv. AN UzSSR, fiz.-matem. nauki, No. 4 (1965).
\({}^{6}\) D. A. Burgess, Proc. Lond. Math. Soc., 13, No. 51, 524 (1963).
\({}^{7}\) O. Saparniyazov, Izv. AN UzSSR, fiz.-matem. nauki, No. 4 (1964).

* In paper \({}^{7}\) there are many inaccuracies.