Abstract Generated abstract
This paper studies density estimates for zeros of Dirichlet L-functions, motivated by Bombieri’s refinement of Linnik’s large sieve method and its applications to primes in arithmetic progressions and additive number theory. It proves two strengthened average zero-density inequalities for primitive characters, using Gabriel’s method together with an eighth-moment estimate on the critical line, and a classical argument relying on Burgess’s bound. As a consequence, it obtains a zero-free region for L-functions attached to almost all moduli and applies this to asymptotic formulas with power-saving error terms for moments of class numbers of imaginary quadratic fields. The paper also formulates a stronger density hypothesis and notes its implications for near-critical zero-free regions and improved class-number moment estimates.
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UDC 511.28+511.43
MATHEMATICS
M. B. BARBAN
ON THE DENSITY HYPOTHESIS OF E. BOMBIERI
(Presented by Academician Yu. V. Linnik on 9 IV 1966)
In a recent work \((^1)\), Bombieri, generalizing Yu. V. Linnik’s “large sieve” method \((^2)\), proved an elegant density theorem, a simple consequence of which is the following form of the “averaged” law of distribution of prime numbers in arithmetic progressions:
\[ \sum_{D\le x^{1/2}/(\ln x)^B}\ \max_{(l,D)=1} \left|\pi(x,D,L)-\frac{\operatorname{li}x}{\varphi(D)}\right| =O\left(\frac{x}{\ln^A x}\right) \tag{1} \]
for any arbitrarily large but fixed \(A\), and for some \(B=B(A)\). Estimate (1) replaces the extended Riemann hypothesis in well-known additive problems \((^3,^4)\). (An estimate close to (1) was obtained by A. I. Vinogradov in \((^5)\), by another method and independently; there an interesting application to the generalized divisor problem of Hooley–Linnik is also given.)
Many problems of this type were first solved by Yu. V. Linnik’s dispersion method \((^6)\), which in essence uses the achievements of modern algebraic geometry and whose natural sphere of action is formed by binary problems not derivable from the extended Riemann hypothesis, for example,
\[ N=Q(x,y)+P(p_1,p_2), \]
where \(Q(x,y)\), \(P(x,y)\) are positive definite quadratic forms; \(p_1,p_2\) are prime numbers, \(p_1\le N^\alpha,\ p_2\le N^{1-\alpha}\).
In a somewhat roughened form Bombieri’s density theorem has the form
\[ \sum_{D\le X}\sum_{\chi_D'} N(a,T,\chi_D') \ll X^{8(1-\alpha)/(3-2\alpha)+\varepsilon}T^c, \tag{2} \]
where \(N(a,T,\chi)\) denotes the number of zeros of the Dirichlet \(L\)-series \(L(s,\chi)\) in the rectangle \(a\le \sigma\le 1,\ |t|\le T\), with \(s=\sigma+it,\ T\ge 2,\ a\ge 1/2\); \(\varepsilon\) is arbitrarily small but fixed; \(\chi_D'\) is a primitive character \(\bmod D\).
In the same work \((^1)\), Bombieri formulates the hypothesis
\[ \sum_{D\le X}\sum_{\chi_D'} N(a,T,\chi_D') \ll X^{4(1-\alpha)+\varepsilon}T^{1+\varepsilon}. \tag{3} \]
Results of type (2), (3) obviously make it possible to conclude that, for almost all moduli \(D\), the quasi-Riemann hypothesis holds (for all \(\chi_D'\), \(L(s,\chi_D')\) have no zeros in the rectangle \(\sigma>\gamma_0,\ |t|\le D^\varepsilon\), \(\gamma_0<1\) an absolute constant). This, in turn, makes it possible to obtain a power saving in the asymptotic formula for the moments of the number of ideal classes of a quadratic field.
The following two inequalities constitute a strengthening of Bombieri’s estimate (2), useful for the applications described.
Theorem 1. The inequalities hold
\[ \sum_{D\le X}\sum_{\chi_D'} N(a,T,\chi_D') \ll X^{10(1-\alpha)/(3-\alpha)+\varepsilon}T^c, \tag{4} \]
\[ \sum_{D \le X} \sum_{\chi_D'} N(\alpha, T, \chi_D') \ll X^{10/4(1-\alpha)+\varepsilon} T^c \tag{5} \]
uniformly for all \(\alpha,\ 1/2 \le \alpha \le 1\).
Inequality (4) is obtained by Gabriel’s method (see (7), p. 238) and the following estimate of the eighth moment of \(L\)-series on the critical line:
\[ \sum_{D \le X} \sum_{\chi_D \ne \chi_0} |L(1/2+it,\chi)|^8 \ll X^{2+\varepsilon} (|t|+2)^c \tag{6} \]
(an obvious consequence of Theorem 3 of Bombieri’s paper (1) and the truncated functional equation for Dirichlet \(L\)-series).
Inequality (5) is proved according to the classical scheme (see (8)). Here the main role is played by Burgess’s estimate (9) for the modulus of the \(L\)-series on the line \(\sigma=1/2\).
An obvious consequence of inequality (4) of Theorem 1 is
Theorem 2. For all moduli \(D\) in the interval \([1,X]\), with the exception of at most \(X^{1-\delta}\), all \(L\)-series corresponding to primitive characters \(\bmod D\) have no zeros in the region \(|\sigma|>7/9+\delta,\ |t|\le D^\gamma\) \((\varepsilon=\varepsilon(\delta,\gamma))\).
Using Theorem 2 according to the scheme of paper (10), we obtain the following theorem.
Theorem 3. Let \(h(-D)\) denote the number of classes of purely radical quadratic forms of determinant \(-D,\ D>0\). Then for any fixed \(k\) one has
\[ \sum_{D \le N} h^k(-D)= \frac{2^{k+1} r(k)}{\pi^k(k+2)} N^{(k+2)/2} \{1+O(N^{-\xi})\}, \tag{7} \]
where
\[ r(k)= \sum_{\substack{n=1\\ n\equiv 1\ (2)}}^{\infty} \frac{\varphi(n)\tau_k(n^2)}{n^3}, \qquad \xi \text{ is any constant smaller than } \]
\[ (\sqrt{129}-9)/2 \approx 0.18. \]
A substantial advance in this question would now be a proof of (7) with \(\xi=1/2+\varepsilon\) (Bombieri’s hypothesis (3) gives \(\xi=1/5+\varepsilon\)).
In conclusion we formulate the hypothesis.
Hypothesis. The inequality
\[ \sum_{D \le X} \max_{\chi_D'} N(\alpha,T,\chi_D') \ll X^{2(1-\alpha)+\varepsilon}T^c \]
holds uniformly for \(1/2 \le \alpha \le 1\).
Let us note that a consequence of this hypothesis is the absence (for almost all moduli \(D\)) of zeros of the functions \(L(s,\chi_D')\) in the rectangle \(\sigma>1/2+\varepsilon;\ |t|\le D^\gamma\), as well as the possibility of choosing \(\xi=1/3+\varepsilon\) in Theorem 3.
Institute of Mathematics
named after V. I. Romanovskii
Academy of Sciences of the Uzbek SSR
Received
5 IV 1966
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