# 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} }

## 36 Citations

Explicit bounds on the logarithmic derivative and the reciprocal of the Riemann zeta-function

- Mathematics
- 2015

: The purpose of this article is consider | ζ (cid:48) ( σ + it ) /ζ ( σ + it ) | and | ζ ( σ + it ) | − 1 when σ is close to unity. We prove that | ζ (cid:48) ( σ + it ) /ζ ( σ + it ) | (cid:54)…

EXPLICIT ZERO‐FREE REGIONS FOR DIRICHLET ‐FUNCTIONS

- Mathematics
- 2018

Let L(s, χ) be the Dirichlet L-function associated to a non-principal primitive character χ modulo q with 3 ≤ q ≤ 400 000. We prove a new explicit zero-free region for L(s, χ): L(s, χ) does not…

Explicit estimates of some functions over primes

- Mathematics
- 2018

New results have been found about the Riemann hypothesis. In particular, we noticed an extension of zero-free region and a more accurate location of zeros in the critical strip. The Riemann…

Density results for the zeros of zeta applied to the error term in the prime number theorem

- Mathematics
- 2022

. We improve the unconditional explicit bounds for the error term in the prime counting function ψ ( x ). In particular, we prove that, for all x > 2, we have | ψ ( x ) − x | < 9 . 22106 x (log x ) 3…

Some explicit estimates for the error term in the prime number theorem

- Mathematics
- 2022

. By combining and improving recent techniques and results, we provide explicit estimates for the error terms | π ( x ) − li( x ) | , | θ ( x ) − x | and | ψ ( x ) − x | appearing in the prime number…

Prime Numbers in Short Intervals

- Mathematics
- 2017

The Riemann zeta function, ζ(s), is central to number theory and our understanding of the distribution of the prime numbers. This thesis presents some of the known results in this area before…

Explicit interval estimates for prime numbers

- MathematicsMath. Comput.
- 2022

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta, x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the…

Explicit lower bounds on $|L(1, \chi)|$

- Mathematics
- 2021

Let χ denote a primitive, non-quadratic Dirichlet character with conductor q, and let L(s, χ) denote its associated Dirichlet L-function. We show that |L(1, χ)| ≥ 1/(9.12255 log(q/π)) for…

Sharper bounds for the error term in the Prime Number Theorem

- Mathematics, Computer Science
- 2022

This work proves that it is proved that π ( x) − Li ( x ) | ≤ 9 .

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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…

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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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An improved estimate is given for $$|\theta (x) -x|$$|θ(x)-x|, where $$\theta (x) = \sum _{p\le x} \log p$$θ(x)=∑p≤xlogp. Four applications are given: the first to arithmetic progressions that have…

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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…

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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.