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Search: MSC category 30C15 ( Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} )

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1. CMB Online first

Maier, Helmut; Rassias, Michael Th.
On the size of an expression in the Nyman-Beurling-Báez-Duarte criterion for the Riemann Hypothesis
A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty \left|1-\zeta A_N \left(\frac{1}{2}+it \right) \right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] where the infimum is over all Dirichlet polynomials $$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$ of length $N$. In this paper we investigate $d_N^2$ under the assumption that the Riemann zeta function has four non-trivial zeros off the critical line.

Keywords:Riemann hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte criterion
Categories:30C15, 11M26

2. CMB 2006 (vol 49 pp. 438)

Mercer, Idris David
Unimodular Roots of\\ Special Littlewood Polynomials
We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if $n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

Categories:26C10, 30C15, 42A05

3. CMB 2000 (vol 43 pp. 105)

Overholt, Marius
Sets of Uniqueness for Univalent Functions
We observe that any set of uniqueness for the Dirichlet space $\cD$ is a set of uniqueness for the class $S$ of normalized univalent holomorphic functions.

Categories:30C55, 30C15

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