# Talk:Generalized Riemann hypothesis

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Field:  Number theory

"The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis."

Such a χ is not a Dirichlet character (as there is no positive k s.t. χ(n)=0 whenever gcd(n,k)>1). Or am I missing something? -- EJ 14:40, 19 Nov 2004 (UTC)

Try k = 1. Algebraist 20:59, 5 Jun 2005 (UTC)

Uhh, yes, thanks. Stupid me. -- EJ 12:51, 11 July 2005 (UTC)

A GRH (disambiguation) page is needed. -Emiellaiendiay 15:23, 5 December 2006 (UTC)

## Any competing hypothesis?

Any competing hypothesis which shows any alternative to GRH?

Reddwarf2956 (talk) 19:01, 25 October 2009 (UTC)

Is there anything known about the asymptotics of π(x,a,d), beyond Dirichlet's result, without assuming GRH? It would seem strange to me if there weren't. If there is a known stronger result, then it should be in the article, to give a clearer sense of what improvement GRH would actually yield. I'm not strong in number theory so I wouldn't know the best place to look. -- Spireguy (talk) 01:37, 31 October 2010 (UTC)

## Is this a mistake?

If GRH is true, then every proper subgroup of the multiplicative group ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}$ omits a number less than ${\displaystyle 2(\ln p)^{2}\,}$

Seems that ${\displaystyle 2(\ln p)^{2}}$ should be ${\displaystyle 2(\ln n)^{2}}$ — Preceding unsigned comment added by 181.28.138.163 (talk) 17:04, 25 November 2011 (UTC)

Fixed.—Emil J. 11:21, 5 April 2012 (UTC)

It now says:

If GRH is true, then every proper subgroup of the multiplicative group ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}$ omits a number less than 2(ln n)2, as well as a number coprime to n less than 3(ln n)2.

Isn't the second statement a direct consequence of the first? If I find a number less than 2(ln n)2 it will surely be less than 3(ln n)2. So what's the point here? --Jobu0101 (talk) 10:26, 24 November 2015 (UTC)