We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.

In categorical data, it is typically the case that some combinations of variables are theoretically impossible, such as a three year old child who is married or a man who is pregnant. In practice, however, reported values often include such structural zeros due to, for example, respondent mistakes or data processing errors. To purge data of such errors, many statistical organizations use a proc...

In this paper D-optimal designs for the weighted polynomial regresŽ 2 . n sion model of degree p with efficiency function 1 x are presented. Interest in these designs stems from the fact that they are equivalent to locally D-optimal designs for inverse quadratic polynomial models. For the unrestricted design space and p n, the D-optimal designs put equal masses on p 1 points which coincide with...

The characteristic equation of a system of delay diierential equations (DDEs) is a nonlinear equation with innnitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e. stability determining, zeros of the characteristic equation. Th...

We study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function, which was observed by a number of authors. We establish a precise measure which explains the phenomenon, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general L-func...

We study so-called real zeros of holomorphic Hecke cusp forms, that is zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed that existence of many such zeros follows if many short intervals contain numbers whose all prime factors belong to a certain subset of the primes. We prove new r...

and Applied Analysis 3 real and there is a need to locate their position. Moreover, the usual methods M1 – M3 mentioned before for the study of the zeros of Pn x may not apply at all, when Pn x are complex, or they may need serious modifications. Instead, the M4 method can be used directly. Such a functional analytic methodwas introduced in 10 andwas successfully used in a series of papers by t...

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...

Linear operators which (1) preserve the reality of zeros of polynomials having only real zeros and (2) map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain Fox–Wright functions and generalized Mittag-Leffler functions. The paper includes several open problems and questions.  2005 Elsevier Inc. All rights reserved.

Based on work of Alain Connes, I have constructed a spectral interpretation for zeros of L-functions. Here we specialise this construction to the Riemann ζ-function. We construct an operator on a nuclear Fréchet space whose spectrum is the set of non-trivial zeros of ζ. We exhibit the explicit formula for the zeros of the Riemann ζ-function as a character formula.