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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?

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I would suggest this for community wiki. – efq 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


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:

which is Chapter 5 of a book "The Riemann hypothesis : a resource for the afficionado and virtuoso alike", by Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller, published by the Canadian Mathematical Society.

In the reference above, they use GRH for Dirichlet L-series (section 6.2), and ERH for Dedekind zeta functions (section 6.5). However, to make things more complicated:

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.

2) They mention that the "Grand Riemann hypothesis" (which I had never heard of) refers to L functions of automorphic cuspidal representations.


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