I wanted to prepare a presentation for the students I tutor on the Clay Millennium problems.

This is directed at the Riemann Hypothesis and the Generalized Riemann Hypothesis.

The Wikipedia article is good at showing how many conjectures be come true if RH or GRH is proven to be true:

  • $GRH \implies Conjecture\;X$.

What am looking for is list of possible solution paths where; if something is true then the GRH (or the RH) is true or something is equivalent to GRH (or the RH):

  • $Conjecture\;X \;\implies GRH$.

  • $Conjecture\;X \iff GRH$.

The Wikipedia article is a good reference for the first bullet point, but not for the last two.

I am looking for references to fill out a list regarding the last two bullet points for my presentation.

  • $\begingroup$ So on the equivalent list there is Li's criterion and Weil's criterion. I did not think this question would be so interesting. I may have to make a page on my domain to capture the results in a centralized location. $\endgroup$ Commented Jan 14, 2014 at 20:19
  • $\begingroup$ I certainly would appreciate the effort of anyone who would collate and maintain such an implication & equivalence list! $\endgroup$ Commented Jul 22, 2015 at 19:37

2 Answers 2


I looked through the wikipedia link, it is also weak on statements equivalent to RH.

The three elementary ones all go back to NICOLAS, see item number 87 at PAPERs which has a pdf and is Jean-Louis Nicolas. Petites valeurs de la fonction d'Euler, J. Number Theory, 17, 1983, 375--388. petitsphi83.pdf

See also PLANAT for a fascinating relationship with Cramer's conjecture. And Is the Euler phi function bounded below?

Next is Robin 1984 . Robin was a student of Nicolas. Evidently I got this article from the library.


These are the ones I would choose to tell students, especially Nicolas. I did a pretty substantial computation of the relevant estimates using the primorial numbers. The relevant column just kept growing, but Planat showed that if it grows forever (the limit is 1) there would be trouble.

Also see any answers I gave with the words Highly Composite Numbers or Colossally Abundant Numbers.

http://en.wikipedia.org/wiki/Divisor_function seems good, also http://en.wikipedia.org/wiki/Highly_composite_number and http://en.wikipedia.org/wiki/Superior_highly_composite_number and http://en.wikipedia.org/wiki/Abundant_number and http://en.wikipedia.org/wiki/Colossally_abundant_number

Planat numbers:

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  • $\begingroup$ Xian-Jin Li has one:en.m.wikipedia.org/wiki/Li's_criterion $\endgroup$ Commented Jan 12, 2014 at 20:28
  • $\begingroup$ @BrianRushton, thank you. Did not know that one. $\endgroup$
    – Will Jagy
    Commented Jan 12, 2014 at 20:31
  • $\begingroup$ Granville & Tucker seems to imply a respectively weaker(?) GRH from ABC. $\endgroup$ Commented Jan 17, 2014 at 7:19
  • $\begingroup$ @BalarkaSen, found it, It's As Easy As $abc,$ Notices of the AMS, volume 49, number 10, November 2002, pages 1224-1231. $\endgroup$
    – Will Jagy
    Commented Jan 17, 2014 at 15:56
  • $\begingroup$ Yes, that is the one. $\endgroup$ Commented Jan 17, 2014 at 15:58

The best I have found so far is: The Riemann Hypothesis A Resource for the Afficionado and Virtuoso Alike.

The is a whole chapter on equivalences.


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