# Equivalent to Riemann Hypothesis

Through last number theory, I did learn that Riemann hypothesis is equivalent to the following inequality : $|\pi(x)-Li(x)| \leq \sqrt{x} log(x)$ where $Li(x)$ is the Logarithmic integral function and $\pi(x)$ is the prime-counting function. So, I have been more interested in it, i'd like to know more conditions which are equivalent to Riemann Hypothesis

Please, Let me know that.

• Do you know how to prove it is equivalent, or at least implies RH ? See this question and this Collection of equivalent forms of Riemann Hypothesis on mathoverflow Commented Sep 6, 2016 at 22:42
• I don't know if in previous collections (those cited by users in comments and answer) were some due to Sondow, van de Lune and Nazardonyavi. You are welcome if in next seasons you need I write such references. I believe that the more important is the genuine complex version itself, learn the relationship between primes and non-trivial zeros. It is a problem of cross-cutting nature (a time ago I've heard in a talk, that is a transalgebraic issue). Also there are questions in the interface between mathematics and physics. All previous my claims are from a divulgative viewpoint.
– user243301
Commented Sep 8, 2016 at 17:14

## 1 Answer

Online lists of equivalents of the Riemann hypothesis include:

http://aimath.org/WWN/rh/ (section C, "Equivalences to RH")

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/RHreformulations.htm

http://en.wikipedia.org/wiki/Riemann_hypothesis#Criteria_equivalent_to_the_Riemann_hypothesis