A. S. FAINLEIB

ON A LIMIT THEOREM FOR THE NUMBER OF CLASSES OF PURELY RADICAL QUADRATIC FORMS OF NEGATIVE DETERMINANT

(Presented by Academician Yu. V. Linnik on 12 VI 1968)

The limit theorem for the function \(h(-D)\) (the number of classes of purely radical quadratic forms of negative determinant \(-D\)) was derived by M. B. Barban \((^{1,2})\) from asymptotic formulas obtained by him for the moments of this function.

A generalization of Esséen’s inequality, recently found by the author \((^{3})\), makes it possible in the present note to prove the following refinement of the indicated limit theorem.

Theorem. For \(x>0,\ N\to\infty\),

\[ \frac{1}{N}\sum_{\substack{D\le N\\ h(-D)\le \frac{2}{\pi}\sqrt{Dx}}} 1 = \Phi(x)+O\left( \frac{\ln\ln N\,(\ln\ln\ln N)^{1/2}}{(\ln N)^{1/2}} \right) \tag{1} \]

uniformly in \(x\), where \(\Phi(x)=F(\ln x)\); \(F(u)\) is the distribution function defined by the characteristic function

\[ f(\xi)=\prod_{p>2}\left(\frac{1}{p}+\frac{1}{2}\left(1-\frac{1}{p}\right) \left\{\left(1-\frac{1}{p}\right)^{-i\xi}+\left(1+\frac{1}{p}\right)^{-i\xi}\right\}\right). \tag{2} \]

We shall precede the proof by several lemmas.

Lemma 1. Let \(\chi_D(n)=(-D/n)\) be the Jacobi symbol, if \((n,2D)=1\); \(\chi_D(n)=0\) otherwise. As \(N\to\infty\),

\[ \frac{1}{N}\sum_{D\le N} L^{i\xi}(1,\chi_D) = f(\xi)+O\left(\exp\left\{-a\frac{\ln N}{\ln\ln N}+b|\xi|\ln\ln\ln N\right\}\right) \tag{3} \]

uniformly in \(\xi\) for all real \(\xi\). Here \(f(\xi)\) is defined by formula (2), and \(a\) and \(b\) are positive constants.

The proof follows the scheme of work \((^{1})\), with a small modification connected with nonintegral exponents and the uniformity of the estimate in \(\xi\). In doing so one uses the density theorem of E. Bombieri \((^{4})\), a consequence of which, as M. B. Barban showed \((^{5})\), is the absence of zeros of Dirichlet \(L\)-series corresponding to primitive characters, for “almost all” moduli in the region \(\sigma\ge 1-\alpha,\ |t|\le k^\beta\), where \(\alpha\) and \(\beta\) are positive constants.

An immediate consequence of Lemma 1 is the limit theorem for \(L(1,\chi_D)\).

Lemma 2. Let \(F(x)\) and \(G(x)\) be distribution functions, and let \(f(\xi)\) and \(g(\xi)\) be their characteristic functions. For \(T>0\),

\[ \sup_x |F(x)-G(x)|<C_1\left\{Q_F\left(\frac{1}{T}\right)+ \int_0^T \left|\frac{f(\xi)-g(\xi)}{\xi}\right|\,d\xi\right\}, \tag{4} \]

where \(C_1\) is an absolute constant, \(Q_F(l)=\sup_x\{F(x+l)-F(x)\}\).

The proof is given in paper \((^3)\).

Lemma 3. Let \(\eta_1,\eta_2,\ldots,\eta_n\) be independent random variables,
\[ \eta=\eta_1+\eta_2+\cdots+\eta_n,\qquad Q_m(l)=\sup_x P(x<\eta_m\le x+l). \]

Then
\[ Q(l)=\sup_x P(x<\eta\le x+l)<C_2\left[\sum_{m=1}^n(1-Q_m(l))\right]^{-1/2}, \tag{5} \]
where \(C_2\) is an absolute constant.

This inequality for concentration functions is due to B. A. Rogozin \((^6)\).

We can now prove the theorem formulated at the beginning. From formula (2) it follows that \(F(x)\) is the distribution function of the sum of independent random variables \(\eta_p\) (\(p\) runs through the odd primes), taking the three values \(0\), \(\ln(1-1/p)^{-1}\), and \(\ln(1+1/p)^{-1}\) with probabilities respectively \(1/p\), \(\frac12(1-1/p)\), and \(\frac12(1-1/p)\). Hence, by Lemma 3,
\[ Q_F(l)=O\left(\sqrt{\,l\ln\frac1l\,}\right). \tag{6} \]

We apply Lemma 2, using, for the estimate of the difference of characteristic functions in a neighborhood of zero, the known estimate \(\ln L(1,\chi_D)=O(\ln D)\), and outside this neighborhood—Lemma 1. We obtain, for \(\delta>0\):
\[ \sup_x\left|F(x)-\frac1N \sum_{\substack{D\le N\\ \ln L(1,\chi_D)<x}}1\right| \ll \sqrt{\frac{\ln T}{T}}+\delta\ln N+ \]
\[ +\ln\frac{T}{\delta}\exp\left\{-a\frac{\ln N}{\ln\ln N}+bT\ln\ln\ln N\right\}. \]

The proof is completed by choosing
\[ \delta=1/\ln^2 N,\qquad T=a\ln N/2b\ln\ln N\cdot \ln\ln\ln N, \]
if one takes into account Gauss’s well-known formula
\[ h(-D)=\frac2\pi \sqrt{D}\,L(1,\chi_D) \]
for \(D>1\).

The author expresses gratitude to M. B. Barban for valuable comments.

Received
30 V 1968

REFERENCES

\(^1\) M. B. Barban, Izv. AN SSSR, Ser. Matem., 26, 4, 573 (1962).
\(^2\) M. B. Barban, UMN, 21, 1 (127), 51 (1966).
\(^3\) A. S. Fainleib, Izv. AN SSSR, Ser. Matem., 32, 4 (1968).
\(^4\) E. Bombieri, Mathematica, 12, 201 (1965).
\(^5\) M. B. Barban, DAN, 172, No. 5, 999 (1967).
\(^6\) B. A. Rogozin, Teoriya veroyatnostei i ee primeneniya, 6, 1, 103 (1961).