Mark Kac gave an explicit formula for the expectation of the number, νn(Ω), of zeros of a random polynomial, Pn(z) = n ∑ j=0 ηjz j , in any measurable subset Ω of the reals. Here, η0, . . . , ηn are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function gn for which Eνn(Ω) = ∫ Ω gn(x)dx. Inspired by that result, Larry Shepp and I found ...

The zeros of the partition function of the ferromagnetic q-state Potts model with long-range interactions in the complex-q plane are studied in the mean-field case, while preliminary numerical results are reported for the finite 1d chains with powerlaw decaying interactions. In both cases, at any fixed temperature, the zeros lie on the arc-shaped contours, which cross the positive real axis at ...

the aim of this note is to characterize the finite groups in which all non-linear irreducible characters have distinct zero entries number.

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among {α1, . . . , αn} ⊂ (C0 ∪ {∞}), are not all real (unless αn is real), and hence, they are not suitable to construct a rational Gaussian quadrature rule (RGQ). For this reason, the zeros of a so-called quasi-ORF or a so-called paraORF are used instead. These zeros depend on one single par...

In this paper, under a weakened version of Hardy—Littlewood Conjecture on the number representations in Goldbach problem, we shall prove bounds for Siegel zeros real primitive Dirichlet characters composite moduli.