1. Introduction. The object of this paper is to investigate the distribution of the nontrivial zeros, near the real axis, of quadratic L-functions.Interest in the zeros of Dirichlet L-functions close to the real axis is well documented. For example, see [1]-[3], [7], [9]- [11]. For instance, if χ is an odd quadratic character, then the zeros of L(s, χ) close to s = 1/2 influence the size of the class number of the complex quadratic number field associated with χ. Also, if χ is the nonprincipal character modulo 4, then the location of the "first" zero of L(s, χ) in the critical strip has a bearing on how the primes are distributed in the residue classes 1 and 3 mod 4, respectively, and in particular, on a phenomenon first observed by Chebyshev [4] concerning the discrepancies in the distribution of primes in different residue classes. Roughly speaking, there are "more" primes congruent to 3 mod 4 than congruent to 1 mod 4.In this paper, let K(s) be a complex function such that |K(1/2 + it)| is a rapidly decreasing function of t > 0. Moreover, let a(x) be its Mellin transform as defined in the next section. Define

We introduce a new number-theoretic spin chain and explore its thermodynamics and connections with number theory. The energy of each spin configuration is defined in a translation-invariant manner in terms of the Farey fractions, and is also expressed using Pauli matrices. We prove that the free energy exists and exhibits a unique phase transition at inverse temperature beta = 2. The free energy is the same as that of a related, non translation-invariant number-theoretic spin chain. Using a number-theoretic argument, the low-temperature (beta > 3) state is shown to be completely magnetized for long chains. The number of states of energy E = log(n) summed over chain length is expressed in terms of a restricted divisor problem. We conjecture that its asymptotic form is (n log n), consistent with the phase transition at beta = 2, and suggesting a possible connection with the Riemann zeta function. The spin interaction coefficients include all even many-body terms and are translation invariant. Computer results indicate that all the interaction coefficients, except the constant term, are ferromagnetic.Comment: 15 pages + 5 figures, postscript. Contact: kleban@maine.ed

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We study the distribution of the imaginary parts of zeros near the real axis of quadratic L-functions. More precisely, let K(s) be chosen so that \K(l/2 ± ii)\ is rapidly decreasing as i increases. We investigate the asymptotic behaviour of

To study the distribution of pairs of zeros of the Riemann zeta‐function, Montgomery introduced the function Ffalse(αfalse)=FTfalse(αfalse)=(T2πlog⁡T)−1∑0<γ,γ′⩽TTiαfalse(γ−γ′false)wfalse(γ−γ′false), where α is real and T ⩾ 2, γ and γ′ denote the imaginary parts of zeros of the Riemann zeta‐function, and w(u) = 4/(4 + u2). Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for F(α) when |α| ⩽ 1, and made the conjecture that F(α) = 1 + o(1) as T → ∞ for any bounded α with |α| ⩾ 1. In this paper we use an approximation for the prime indicator function together with a new mean value theorem for long Dirichlet polynomials and tails of Dirichlet series to prove that, assuming the Generalized Riemann Hypothesis for all Dirichlet L‐functions, then for any ε > 0 we have Ffalse(αfalse)⩾32−|α|−ϵ, uniformly for 1⩽|α|⩽32−2ϵ and all T ⩾ T0(ε). 1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.