# Riemann hypothesis generalization names: extended versus generalized?

This is a "names" question. There are two standard directions of generalization of the Riemann hypothesis: one to L-functions (which is used quite a bit in analytic number theory, and for extending density results for primes to Dirichlet primes) and another to Dedekind zeta-functions. Wikipedia says that the version for L-functions is called the GRH and the version of zeta-functions is called the ERH, and other generalizations are called the GRH. But I have seen conflicting terminology in some books and papers, which refer to the L-function version as the ERH and the other version as the GRH.

Which is the more standard convention?

• I would suggest this for community wiki. – M.G. Feb 9 '10 at 20:31
• Until someone convinces me otherwise I am going to go with the assertion that "zeta function" and "L-function" mean the same thing, and the only reason that one is sometimes preferred over the other is historical coincidence. – Kevin Buzzard Feb 9 '10 at 20:46
• @Kevin: An L-function is what you get by starting with a zeta function and adding a representation, no? – Pete L. Clark Feb 9 '10 at 21:14

Hello,

I also agree that the literature is not quite consistent on this topic. I tried to find a published reference on this question and found the following article:

1) They mention (in section 6.3) that ERH may be referring to the conjecture for L series of the form $$\sum_{n=1}^\infty \frac{\left(\frac{n}{p}\right)}{n^s}$$ where $p$ is a prime and $\left(\frac{n}{p}\right)$ is the Legendre symbol. In fact they call this version ERH, and the one for Dedekind zeta functions is called another extended Riemann hypothesis.