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Search: All articles in the CMB digital archive with keyword Riemann Hypothesis

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1. CMB Online first

Maier, Helmut; Rassias, Michael Th.
 On the size of an expression in the Nyman-Beurling-BÃ¡ez-Duarte criterion for the Riemann Hypothesis A crucial role in the Nyman-Beurling-BÃ¡ez-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-BÃ¡ez-Duarte criterionCategories:30C15, 11M26

2. CMB 2011 (vol 54 pp. 316)

Mazhouda, Kamel
 The Saddle-Point Method and the Li Coefficients In this paper, we apply the saddle-point method in conjunction with the theory of the NÃ¶rlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have $$\lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n),$$ with $$c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},$$ where $\gamma$ is the Euler's constant and the notation is as below. Keywords:Selberg class, Saddle-point method, Riemann Hypothesis, Li's criterionCategories:11M41, 11M06
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