Let G(z) be an entire function of order less than 2 that is real for real z with only real zeros. Then we classify certain distribution functions F such that the convolution (G ∗ dF )(z) = ∫∞ −∞ G(z − is) dF (s) of G with the measure dF has only real zeros all of which are simple. This generalizes a method used by Pólya to study the Riemann zeta function.

Motivation: An important feature of microbiome count data is the presence of a large number of zeros. A common strategy to handle these excess zeros is to add a small number called pseudo-count (e.g., 1). Other strategies include using various probability models to model the excess zero counts. Although adding a pseudo-count is simple and widely used, as demonstrated in this paper, it is not id...

The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 2 (ξ(s + h) + ξ(s− h)) and Bh(s) = 1 2i (ξ(s+ h)− ξ(s− h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line R(s) = 1 2 and are simple zeros. The number of zeros of these functions ...

We state precise results on the complexity of a classical bisectionexclusion method to locate zeros of univariate analytic functions contained in a square. The output of this algorithm is a list of squares containing all the zeros. It is also a robust method to locate clusters of zeros. We show that the global complexity depends on the following quantities: the size of the square, the desired p...

The Riemann zeta function is given by: [equation, see published text]. Zeta(s) may be analytically continued to the entire s-plane, except for a simple pole at s = 0. Of great interest are the complex zeros of zeta(s). The Riemann hypothesis states that the complex zeros all have real part 1/2. According to the prime number theorem, pn approximately n logn, where pn is the nth prime. Suppose th...

Abstract. We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-a...

Characteristic polynomials of unitary matrices are extremely useful models for the Riemann zeta-function ζ(s). The distribution of their eigenvalues give insight into the distribution of zeros of the Riemann zeta-function and the values of these characteristic polynomials give a model for the value distribution of ζ(s). See the works [KS] and [CFKRS] for detailed descriptions of how these model...

We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

Use of a common-acoustical-pole and zero model is proposed for modeling head-related transfer functions (HRTF’s) for various directions of sound incidence. The HRTF’s are expressed using the common acoustical poles, which do not depend on the source directions, and the zeros, which do. The common acoustical poles are estimated as they are common to HRTF’s for various source directions; the esti...