Explicit estimates of some functions over primes

@article{Dusart2018ExplicitEO,
  title={Explicit estimates of some functions over primes},
  author={Pierre Dusart},
  journal={The Ramanujan Journal},
  year={2018},
  volume={45},
  pages={227-251}
}
  • P. Dusart
  • Published 2018
  • Mathematics
  • The Ramanujan Journal
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 hypothesis implies results about the distribution of prime numbers. We get better effective estimates of common number theoretical functions which are closely linked to $$\zeta $$ζ zeros like $$\psi (x),\vartheta (x),\pi (x)$$ψ(x),ϑ(x),π(x), or the $$k\mathrm{{th}}$$kth prime number $$p_k$$pk. 
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References

SHOWING 1-10 OF 35 REFERENCES
A zero density result for the Riemann zeta function
In this article, we prove an explicit bound for $N(\sigma,T)$, the number of zeros of the Riemann zeta function satisfying $\sigma < \Re s <1 $ and $0 < \Im s < T$. This result provides a significantExpand
On the zeros of the Riemann zeta function in the critical strip
We describe a computation which shows that the Riemann zeta function ζ(s) has exactly 75,000,000 zeros of the form σ+ it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on theExpand
Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$
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 thatExpand
On a Constant Related to the Prime Counting Function
Let $${\pi(x)}$$π(x) be the number of primes not exceeding x. We produce new explicit bounds for $${\pi(x)}$$π(x) and we use them to obtain a fine frame for the remainder term in the asymptoticExpand
Estimates of ψ,θ for large values of x without the Riemann hypothesis
  • P. Dusart
  • Mathematics, Computer Science
  • Math. Comput.
  • 2016
TLDR
The quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual Chebyshev prime and prime power counting functions is drawn to find an upper bound function directly linked to a zero-free region. Expand
Updating the error term in the prime number theorem
An improved estimate is given for $$|\theta (x) -x|$$|θ(x)-x|, where $$\theta (x) = \sum _{p\le x} \log p$$θ(x)=∑p≤xlogp. Four applications are given: the first to arithmetic progressions that haveExpand
Short Effective Intervals Containing Primes
We prove that if $x$ is large enough, namely $x\ge x_0$, then there exists a prime between $x(1- \Delta^{-1})$ and $x$, where $\Delta$ is an effective constant computed in terms of $x_0$. ThisExpand
An explicit density estimate for Dirichlet L-series
TLDR
This paper focuses in this paper on estimating the location of the zeroes of these functions in the strip 0 < <s < 1; the Generalized Riemann Hypothesis asserts that all of those are on the line <s = 1/2. Expand
Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function
Abstract We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 − 1 / ( 5.573412 log ⁡ | t | ) for | t | ≥ 2 . This represents the largest known zero-free regionExpand
New bounds for π(x)
TLDR
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. Expand
...
1
2
3
4
...