Total Number Of Zeros Research Articles

Zeros of a family of approximations of Hecke L-functions associated with cusp forms

We consider a family of approximations of a Hecke L-function \(L_f(s)\) attached to a holomorphic cusp form f of positive integral weight k with respect to the full modular group. These families are of the form $$\begin{aligned} L_f(X;s):=\sum _{n\le X}\frac{a(n)}{n^s}+(-1)^{k/2}(2\pi )^{-(1-2s)}\frac{\Gamma \left( \tfrac{k+1}{2}-s\right) }{\Gamma \left( \tfrac{k-1}{2}+s\right) }\sum _{n\le X}\frac{a(n)}{n^{1-s}}, \end{aligned}$$ where \(s=\sigma +it\) is a complex variable and a(n) is a normalized Fourier coefficient of f. From an approximate functional equation, one sees that \(L_f(X;s)\) is a good approximation to \(L_f(s)\) when \(X=t/2\pi \). We obtain vertical strips where most of the zeros of \( L_f(X;s) \) lie. We study the distribution of zeros of \(L_f(X;s)\) when X is independent of t. For \(X=1\) and 2, we prove that all the complex zeros of \(L_f(X;s)\) lie on the critical line \(\sigma =1/2\). We also show that as \(T\rightarrow \infty \) and \( X=T^{o(1)} \), \(100\,\%\) of the complex zeros of \( L_f(X;s) \) up to height T lie on the critical line. Here by \(100\,\%\) we mean that the ratio between the number of simple zeros on the critical line and the total number of zeros up to height T approaches 1 as \(T\rightarrow \infty \).

On locating clusters of zeros of analytic functions

Given an analytic function f and a Jordan curve γ that does not pass through any zero of f, we consider the problem of computing all the zeros of f that lie inside γ, together with their respective multiplicities. Our principal means of obtaining information about the location of these zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along γ. If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its weight, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational interpolation at roots of unity. Numerical examples illustrate the effectiveness of our techniques.

Zeros on the temperature axis of spin correlations in a random bond Ising chain

The authors study the correlation between two spins diametrically opposite in a random Ising loop of 2r spins with random nearest-neighbour couplings. As a function of the inverse temperature beta , this correlation undergoes random sign changes with a density rho r( beta ). They determine this density for an arbitrary distribution P(K) of the coupling constants for (i) r to infinity and (ii) the scaling limit beta to infinity , r to infinity at r/ beta fixed. For r to infinity , the total number of zeros on the beta -axis grows asymptotically as 1/2 1/21 zeta (3)-1/4 pi 21/2 log r. As an application, the density of zeros is calculated for the spin-spin correlation in a double infinite Ising chain with random bonds, having, with a probability p, an infinite transverse coupling between each pair of corresponding sites. A connection is made with predictions from spin glass theory.