# 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}
}
• Published 15 October 2014
• Mathematics
• Journal of Number Theory
36 Citations

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## References

SHOWING 1-10 OF 27 REFERENCES
Zero-free regions for the Riemann zeta function
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
• Mathematics
• 2014
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
Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$
• Mathematics
• 1975
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
Zeros of the Riemann Zeta Function
In this paper, we present a proof of the Riemann hypothesis. We show that zeros of the Riemann zeta function should be on the line with the real value 1/2, in the region where the real part of
New bounds for π(x)
• Mathematics
Math. Comput.
• 2015
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.
Some extremal properties of positive trigonometric polynomials
AbstractFor n=8 an upper bound is given for the functional $$V_n = \mathop {\inf }\limits_{t_n } \frac{{\alpha _1 + \alpha _2 + \cdots + \alpha _n }}{{\left( {\sqrt {\alpha _1 } - \sqrt {\alpha _0 } Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n) We prove that the error term \sum_{n\le x} \Lambda(n)/n − \log x + \gamma differs from (\psi(x) − x)/x by a well controlled function. We deduce very precise numerical results from this formula. Updating the error term in the prime number theorem An improved estimate is given for$$|\theta (x) -x|$$|θ(x)-x|, where$$\theta (x) = \sum _{p\le x} \log pθ(x)=∑p≤xlogp. Four applications are given: the first to arithmetic progressions that have
New bounds for $\psi(x)$
• Mathematics
• 2013
In this article we provide new explicit Chebyshev's bounds for the prime counting function $\psi(x)$. The proof relies on two new arguments: smoothing the prime counting function which allows to
A still sharper region where π(x)-li(x) is positive
• Mathematics
Math. Comput.
• 2015
A stronger version of Lehman’s theorem involving a different weight function is derived, which enables us to certify the preceding candidate region and improve this region by appealing to a combination of theoretical results and numerical computation.