Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function
@article{Mossinghoff2014NonnegativeTP, title={Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function}, author={Michael J. Mossinghoff and Tim Trudgian}, journal={Journal of Number Theory}, year={2014}, volume={157}, pages={329-349}, url={https://api.semanticscholar.org/CorpusID:117968965} }
62 Citations
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Mathematics
We improve existing explicit bounds of Vinogradov-Korobov type for zero-free regions of the Riemann zeta function, both for large height t and for every t. A primary input is an explicit bound of the…
A NOTE ON KADIRI'S EXPLICIT ZERO FREE REGION FOR RIEMANN ZETA FUNCTION
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Mathematics
Abstract. In 2005 Kadiri proved that the Riemann zeta function ζ(s)does not vanish in the regionRe(s) ≥ 1 −1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 5.69693. In this paper we will show that R 0 can be…
ON SOME EXTREMAL PROBLEMS OF LANDAU
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The prime number theorem with error term presents itself as π(x) = x ∫ 2 dt log t + O ( xe log L x ) . In 1909, Edmund Landau provided a systematic analysis of the proof seeking better values of L…
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Abstract : The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before. They give improved methods for using this and a recent determination that…
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Mathematics
This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is…
Some extremal properties of positive trigonometric polynomials
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AbstractA class Pn of even positive trigonometric polynomials tn(ϕ)=a0 + a1 cos ϕ+ ... + an cos · nϕ, satisfying the conditions: ak ≥0 (k = 0,1, ..., n), a0 < a1 is considered. The behavior of the…
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The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function.
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The Riemann Zeta function $\zeta(s)$ never vanishes in the region : $$ \Re s \ge 1- \frac1{5.70176 \log |\Im s|} \quad \quad (|\Im s| \ge 2). $$
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Mathematics
AbstractFor n=8 an upper bound is given for the functional
$$V_n = \mathop {\inf }\limits_{t_n } \frac{{\alpha _1 + \alpha _2 + \cdots + \alpha _n }}{{\left( {\sqrt {\alpha _1 } - \sqrt {\alpha _0 }…
Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)
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We prove that the error term $\sum_{n\le x} \Lambda(n)/n − \log x + \gamma$ differs from $(\psi(x) − x)/x$ by a well controlled function. We deduce very precise numerical results from this formula.