mersenneforum.org  

Go Back   mersenneforum.org > Math Stuff > Other Mathematical Topics

Reply
 
Thread Tools
Old 2021-06-23, 16:29   #1
mart_r
 
mart_r's Avatar
 
Dec 2008
you know...around...

29B16 Posts
Default A probabilistic measure arising in approximating pi(x) / Question

Given the logarithmic integral
\[Li(x) = \int_2^x \frac{dt}{\log t}\]
and the smooth part of Riemann's prime counting formula as the equivalent of the Gram series
\[R(x) = 1+\sum_{n=1}^\infty \frac{\log^n x}{n\cdot n!\cdot\zeta(n+1)},\]
is there a constant c such that
\[c = \lim_{m\rightarrow\infty} \frac{1}{m} \sum_{x=2}^m \log(\lvert\frac{Li(x)-\pi(x)}{R(x)-\pi(x)}\rvert)\]
?
mart_r is offline   Reply With Quote
Old 2021-07-19, 18:48   #2
mart_r
 
mart_r's Avatar
 
Dec 2008
you know...around...

66710 Posts
Default

Let me admit at this point that I still don't seem to understand how to approximate pi(x) by employing the nontrivial zeta zeroes, specifically how I achieve sufficient convergence of x^(1/2 ± t i) as t gets larger.
I'd like to see how well the above mentioned value c fares when x is large, but for this I need to have a pi(x) approximation that makes use of at least a couple of those zeroes.


Are there any freely available programs for this?
mart_r is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
No way to measure record prime gaps Bobby Jacobs Prime Gap Searches 42 2019-02-27 21:54
probabilistic number theory wildrabbitt Math 57 2015-09-17 18:26
On a curius sighting of a pair of math constants in the deutrons as measure Kathegetes Miscellaneous Math 16 2014-07-13 03:48
approximating products only_human Puzzles 0 2010-03-23 18:49
Approximating Factoring Speed hasan4444 Factoring 17 2009-10-28 14:34

All times are UTC. The time now is 03:31.


Sun Jul 25 03:31:44 UTC 2021 up 1 day, 22 hrs, 0 users, load averages: 1.35, 1.60, 1.85

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.