Open Collections

Sharper bounds for Chebyshev's $\theta(x)$ function

Banff International Research Station for Mathematical Innovation and Discovery

In 1792, Gauss conjectured that the primes occur with a density of $\frac{1}{log x}$ around $x$. Therefore, when developing explicit results relating to the Prime Number Theorem, it is useful to study Chebyshevâ à ôs $\theta(x)$ function, given by \[ \sum_{p \le x} \log p .\] Over summer 2017, I worked on a joint project supported by NSERC USRA to develop an effective version of the Prime Number Theorem. In this talk, I present our results which are the current best results for the prime counting function $\theta(x)$ for various ranges of x. We developed these results by first surveying existing explicit results from the past 60 years on prime counting functions. Our results are based on a recent zero density result for the zeroes of the Riemann Zeta function (due to H. Kadiri, A. Lumley, and N. Ng).

Mathematics; Number theory; Group theory and generalizations

video/mp4

Author affiliation: University of Lethbridge

2019-03-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68662

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/