Talk:Special values of L-functions

Leibniz formula a bad example

Saying that the study of special values of L-functions is about generalizing things like the Leibniz formula for π makes it seem like it's about expressing transcendental numbers as infinite series, which I don't think it's about. It's true that you often need transcendental numbers in your formula, but you're just as happy without them. I think it would be better to say that it's about generalizing Euler's formula on the special values of the Riemann zeta function (and its relation to class numbers of cyclotomic fields) and Dirichlet's analytic class number formula, i.e. finding ways to extract the arithmetic information encoded in L-functions. Opinions? RobHar (talk) 19:21, 10 November 2010 (UTC)

Well, OK. It's not a _bad_ example, in the sense that it is a special case of the analytic class number formula. And it's a striking example. A Disappearing Number starts with the formula harmonic series = -1/12, which therefore has star quality. Consider what function it is performing in the article: to give someone with a little background where this topic fits in, or to satisfy purists who would prefer to start from ζ(2n) for high-flown reasons. This is Wikipedia, after all, and we are supposed to write for the "general reader", however often mathematicians disregard this advice. Charles Matthews (talk) 20:15, 10 November 2010 (UTC)

I agree that we should attempt to allow readers with little background to understand where the topic fits in; my contention is that the current example in the lede misleads the reader about where this topic fits in. The current example makes it seem as though this topic fits under attempting to generalize cute formulas for π. Rather this topic starts with arithmetic data that is encoded into an L-function and then attempts to extract arithmetic information from the values of the L-function. Just to be clear, I'm not against the Leibniz formula being the first example, I just don't think that the study of special values is about generalizing the Leibniz formula. It's about interpreting the Leibniz formula. And I think we should find a way of expressing that. RobHar (talk) 21:19, 10 November 2010 (UTC)

Equivalence to the Iwasawa Main Conjecture is unhelpful

It is indeed very probable that the ETNC and the Iwasawa Main Conjecture are equivalent, but only in the weak logical sense that both are believed to be true. In the more usual sense that one could deduce one from the other, then the ETNC implies the Iwasawa Main Conjecture but the converse is certainly not true. One needs many other hard results to get a proof of the ETNC from the Iwasawa Main Conjecture. Therefore, I suggest deleting the last sentence. — Preceding unsigned comment added by 129.175.50.21 (talk) 09:53, 30 September 2016 (UTC)