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 theorem. For example, we ...show for all x≥2 that |ψ(x)−x|≤9.39x(logx)1.515exp(−0.8274logx). Our estimates rely heavily on explicit zero-free regions and zero-density estimates for the Riemann zeta-function, and improve on existing bounds for prime-counting functions for large values of x.
Sharper bounds for the Chebyshev function ψ(x) Fiori, Andrew; Kadiri, Habiba; Swidinsky, Joshua
Journal of mathematical analysis and applications,
11/2023, Volume:
527, Issue:
2
Journal Article
Peer reviewed
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|<9.22022x(logx)3/2exp(−0.8476836logx), and that, for all x≥exp(3000),|ψ(x)−x|<4.9678⋅10−15x. This compares to results of Platt and Trudgian (2021) who obtained 4.51⋅10−13x. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning π(x) will appear in the follow up work 11.
For x≥y>1 and u:=logx/logy, let Φ(x,y) denote the number of positive integers up to x free of prime divisors less than or equal to y. In 1950 de Bruijn 4 studied the approximation of Φ(x,y) by the ...quantityμy(u)eγxlogy∏p≤y(1−1p), where γ=0.5772156... is Euler's constant andμy(u):=∫1uyt−uω(t)dt. He showed that the asymptotic formulaΦ(x,y)=μy(u)eγxlogy∏p≤y(1−1p)+O(xR(y)logy) holds uniformly for all x≥y≥2, where R(y) is a positive decreasing function related to the error estimates in the Prime Number Theorem. In this paper we obtain numerically explicit versions of de Bruijn's result.
Primes between consecutive powers Cully-Hugill, Michaela
Journal of number theory,
June 2023, 2023-06-00, Volume:
247
Journal Article
Peer reviewed
Open access
This paper updates the explicit interval estimate for primes between consecutive powers. It is shown that there is least one prime between n155 and (n+1)155 for all n≥1. This result is in part ...obtained with a new explicit version of Goldston's 1983 estimate for the error in the truncated Riemann–von Mangoldt explicit formula.
What is the number of rolls of fair six-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive ...error of less than
. This is a solution to a puzzle suggested by DasGupta in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes.
PDF
