Abstract Generated abstract
This paper studies values of L(1, χD) for real Dirichlet characters when the modulus D is restricted to sparse arithmetic progressions and related sets. Using estimates for double sums with quadratic Dirichlet characters, it derives a main lemma reducing asymptotics for moments of modified L-values to an average equidistribution condition on the selected sequence of moduli, and applies this to squarefree D and, conditionally through large sieve results, to prime values. The resulting theorems give moment asymptotics and almost-all formulas showing that on progressions with suitably chosen modulus the typical value of L(1, χD) is governed by an explicit Euler product over small primes. Consequences include sharper large and small value asymptotics for L(1, χD) and applications to class numbers, regulators, quadratic forms, and divisor classes of quadratic fields.
Full Text
UDC 511
MATHEMATICS
A. F. LAVRIK
ON \(L(1,\chi)\) WITH A REAL DIRICHLET CHARACTER ON SPARSE SETS OF VALUES OF THE CHARACTER MODULUS
(Presented by Academician I. M. Vinogradov on 10 VII 1969)
The study of the functions
\[ L(1,\chi_D)=\sum_{n=1}^{\infty}\chi(D,n)n^{-1}, \]
where \(\chi(D,n)\) denotes: \(\chi_n(D)=\left(\frac{+D}{n}\right)\), the Jacobi symbol or zero; or \(\chi_D(n)=\left(\frac{+D}{n}\right)\), the Kronecker symbol or zero, was carried out in \((^{1-8})\) and is connected with applications to the theory of quadratic forms and the theory of divisors of quadratic fields. Recently the problem of the distribution of values of \(L(1,\chi_D)\) has acquired additional interest in connection with A. Selberg’s formula \((^9)\) for the trace of Hecke operators in the space of modular forms.
In the present paper new information is given on the nature of the behavior of the functions \(L(1,\chi_D)\), obtained on the basis of the method indicated in \((^{10})\) for estimating double sums with a quadratic Dirichlet character.
Notation: \(D=dm+l,\ 1\le l\le d,\ (l,d)=1,\ m=1,2,\ldots;\ b_D\) is any sequence of zeros and ones; \(\gamma=2\) or \(1\), according as \(\chi(D,n)=\chi_n(D)\) or \(\chi_D(n)\); \(\tau_k(m)\) is the number of solutions of the equation \(m=m_1\ldots m_k\) in integers \(m_\nu\ge 1\),
\[ H_k(N)=\sum_{m=1}^{N}\left[b_D\prod_{p\mid d}\left(1-\frac{\chi(D,p)}{p}\right)L(1,\chi_D)\right]^k, \]
where \(p\) is a prime number.
Main lemma. Uniformly with respect to \(N,d\), and \(l\) we have
\[ H_k(N)= \sum_{\substack{n=1\\(n,\gamma)=1}}^{N} \sum_{\substack{m=1\\(D,n)=1}}^{N} \frac{\tau_k(n^2)\chi(d^2,n)b_D}{n^2} +O\left(N^{9/10}d^{4/5}\ln^c dN\right)+ \]
\[ +O\left(\ln^c dN \left[ \sum_{2\le n\le X} \sum_{\substack{r=1\\(r,n)=1}}^{n} \left| \sum_{\substack{m=1\\D\equiv r(\operatorname{mod} n)}}^{N} b_D-F \right|^2 \right]^{1/2}\right), \]
where \(F\) is an arbitrary quantity not depending on \(l\); \(c=c(k)\); \(X=N^{2/3}d^{3/10}\).
Thus the problem of the asymptotics for the moments of the function
\[ b_D\prod_{p\mid d}\left(1-\frac{\chi(D,p)}{p}\right)L(1,\chi_D) \tag{1} \]
reduces only to the equidistribution of the sequence \(b_D\) “on average” over primitive arithmetic progressions. As is known, a broad class of sequences possesses a property of this kind. In particular, for \(b_D=\mu^2(D)\), where \(\mu\) is the Möbius function, the following is obtained.
Theorem 1. Uniformly with respect to \(N\), \(d\), and \(l\) we have
\[ H_k(N)=N\sum_{\substack{n=1\\(n,\gamma)=1}}^\infty \sum_{r/n}\sum_{\delta=1}^\infty \frac{\tau_k(n^2)\chi(d',n)\mu(r)\mu(\delta)\varphi(r)} {(nr\delta)^2} + O\left(N^{8/9}d^{4/5}\ln^C dN\right), \]
where \(\varphi(r)\) is Euler’s function.
With the aid of this theorem one can obtain rather varied information concerning the distribution of the values of the function \(L(1,\chi_D)\). For example, from it and from the theorem of Fréchet and Shohat \((^{11,12})\) there follows a limit theorem for the distribution of the functions (1) in arithmetic progressions whose difference \(d\leq N^{1/9}\) grows together with \(N\).
As consequences of Theorem 1 one can also obtain information of a different, less usual, nature. Namely, let henceforth \(d\) denote the product of the primes not exceeding \(q\geq 2\). In this case the following holds.
Theorem 2. For an arbitrary integer \(k\geq 1\), uniformly with respect to \(N\), \(d\leq N^{1/9}\), \(1\leq l\leq d\), \((l,d)=1\), we have
\[ \sum_{m=1}^{N}\mu^2(D)L^k(1,\chi_D) = N\prod_{p/d}\left(1-\frac{\chi(l,p)}{p}\right)^{-k} \left[1+O\left(\frac{\ln^2 q}{q}\right)\right]. \]
This theorem, in particular, shows that the quantity \(\mu^2(D)L(1,\chi_D)\) on the progressions \(dm+l\) is, on the average, independent of \(m\) and is determined only by the difference of the progression \(d\) and its initial term \(l\). It is very plausible that in fact the quantity \(L(1,\chi_D)\) on the indicated set of values \(D\) is, in the main, always equal to
\[ \prod_{p/d}\left(1-\frac{\chi(l,p)}{p}\right)^{-1}. \]
In any case, the following is certainly true.
Theorem 3. Under the conditions of Theorem 2, for each \(m=1,2,\ldots,N\) with \(\mu^2(dm+l)=1\), except for at most \(N/\ln^{1-\varepsilon}N\) of them, where \(\varepsilon>0\),
\[ L(1,\chi_{dm+l}) = \prod_{p/d}\left(1-\frac{\chi(l,p)}{p}\right)^{-1} \left[1+O\left(\frac{1}{\ln\ln 2d}\right)\right]. \]
Hence, with the aid of the known Mertens formulas, one obtains asymptotic expressions for the “large” and “small” values of \(L(1,\chi_D)\). Namely, there exist \(l=l'\) and \(l=l''\) such that, for almost all values \(D\) in the sense of Theorem 3, we have
\[ L(1,\chi_D)=e^c\ln\ln D+O(1) \]
and, respectively,
\[ L(1,\chi_D)=\frac{\pi^2}{6e^c\ln\ln D} \left(1+O\left(\frac{1}{\ln\ln D}\right)\right), \]
where \(c\) is Euler’s constant.
These relations constitute a substantial strengthening of the known inequalities \((^{1-8})\) for \(L(1,\chi_D)\), both in the sense of estimating the approximation to the main term of the growth of \(L(1,\chi_D)\), and in the sense of the density of the values \(D\) for which they hold.
If in the basic lemma one takes \(b_D=1\) when \(dm+l\) is a prime number, and \(b_D=0\) otherwise, then from this lemma and from recent results of the “large sieve” in the form of Theorem 8 of \((^{13})\) there will follow entirely analogous results concerning the values of \(L(1,\chi_D)\) on the set of primes \(D=dm+l\).
Applications to the theory of quadratic forms and divisors of quadratic fields consist in a simple combination of the estimates given here with the corresponding exact Dirichlet formulas (14) for the number of classes of non-equivalent quadratic forms of discriminant \(\pm D\) and the number of divisor classes of quadratic fields.
Let, for example, \(h\) and \(R\) denote, respectively, the class number and the regulator of the quadratic field of discriminant \(D\). Then, for almost all values of \(D\) in the sense of Theorem 3, the following asymptotic law holds:
\[ \ln hR=\ln \sqrt{|D|}-\ln \prod_{p/d}\left(1-\frac{\chi(l,p)}{p}\right) -\frac{1+\operatorname{sign} E}{2}\ln 2 -\frac{1-\operatorname{sign} D}{2}\ln \pi +O\left(\frac{1}{\ln\ln 2d}\right). \]
Tashkent Institute
of Railway Transport Engineers
Received
26 VI 1969
REFERENCES
- J. Littlewood, Proc. Lond. Math. Soc., 2 (1928).
- R. Paley, J. Lond. Math. Soc., 3 (1938).
- Yu. V. Linnik, DAN, 37, No. 4 (1942).
- A. Walfish, Trans. Inst. Math. Tbilisi, No. 11 (1942).
- S. Chowla, Proc. Nat. Acad. Sci. India, 13 (1947).
- S. Chowla, Proc. Lond. Math. Soc. (2), 50 (1949).
- P. T. Bateman, S. Chowla, P. Erdös, Publ. Math., 1, Debrecen (1949–1950).
- M. B. Barban, UMN, 21, issue 1 (127) (1966).
- A. Selberg, Sborn. per. Matematika, 1, 4 (1957).
- A. F. Lavrik, DAN, 186, No. 1 (1969).
- M. Frechet, J. Shohat, Trans. Am. Math. Soc. 33 (1931).
- M. B. Barban, Izv. AN SSSR, ser. matem., 26, 4 (1962).
- K. Prachar, Distribution of Prime Numbers, Moscow, 1967.
- Z. Borevich, I. R. Shafarevich, Lectures on Number Theory, Moscow, 1966.