Ramanujan Summation and Divergent series in relation to the Riemann Zeta function.

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  • #26
Hurkyl
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Gee I must've missed that bit..
NOT!
It IS valid for many sorts of generalized functions as is apparent from my 2nd post (though that's not explicitly Ramanujan summation, for more consider the Euler MacLaurin formula)
Wrong. But I'll chalk that up to not knowing what I meant by "generalized function". If the functions [itex]f_n[/itex] are defined by

[tex]f_n(x) := \begin{cases} \frac{1}{2n} & x \in [-n, n] \\ 0 & x \notin [-n, x] \end{cases}[/tex]

Then the limit [itex]\lim_{n \rightarrow 0} f_n[/itex] doesn't exist. But if we instead work in the space of, for example, Schwartz distributions, then the limit exists and is equal to the delta function.

The general procedure here is to expand your universe of discourse so that (ordinary!) limits and sums that failed to converge in the old universe really do converge in the new universe.

This is an entirely different method than what you've been considering. You retain the ordinary definition of "limit", "sum", "integral", et cetera, but you expand the class of objects you're working with.



These new summation operators are most all legitimate, can be made rigorous, consistent
Nobody said otherwise. The point is that they are NEW summation operators, and the class of valid relations involving these NEW summation will be different than the class of valid relations involving the OLD summation operator.


and are most inconveniently for those who curl up in their comfy math beds(or dare I say Couches) - true.
It is all in the definition as Hurkyl apparently concedes and does not invade upon the rigours of pure mathematics(as that is indeed my main area of interest)
Can the attitude right now.



Oh and I noticed you seem to think of the summation operator as uniquely defined
Well duh. The summation operator from elementary calculus has a precise definition that distinguishes it from all other possible operators. The other (rigorous) generalizations of the summation operator I know also have precise definitions that distinguish them from all other possible operators.
 
  • #27
Haelfix
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The milllion dollar physics question is: Given any naively diverging sum appearing in some calculation, which summation device do you use, or even are you allowed to use.

Normally we appeal to some overarching symmetry principle (for instance the 1+2+3 sequence appears in a functional integral calculation of the casimir energy so we kinda are allowed to use dimensional regularization which also yields -1/12). However I have absolutely no idea why zeta regularization is permitted
 
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You might find
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103900982

a helpful link, I for one brushed over some of the physics but Dr Hawkin's argument is straightforward and builds up a strong case for Zeta regularization over the dimensional method in certain instances.

And Hurkyl, you're right about the generalizations- I didn't think you wanted to expand the space you were working on, indeed that sort of thing is more akin to dimensional antics in all sincerity.
Paul Dirac fell into disapproval with Von Neumann when he first used the Dirac Delta function(introduced by Heaviside if I am not mistaken) but Schwarz distribution theory expanded the scope of a function and made it legitimate. Similarly the 'spaces' we work with are simply the generalizations of our devices. The best example, perhaps, is that the representation 1+ z+ z^2+ z^3+ ... = (1-z)^-1 is valid for all z(except perhaps the pole at unity) under the analytical uniqueness afforded by complex function theory. A bigger space yes, and more powerful tools too- but most definitions remain the same, the sum at least, since one can possibly(though not all that successfully) defer for the case of integration. In effect the new sum operators are in themselves 'suitable generalizations' of a 'classical'(though NOT absolute) operator. You may choose to consider them completely different entities and proceed as you say, but that forces an ugly disconnectedness, for even though definitions must stand faultless our notion of sum (indeed any other object we may seek to generalize) has not faltered (except perhaps the feeling of impossibility which I attribute to something more psychological than mathematical and which has prompted this, dare I say heated discussion)
Note how I haven't used any sum that isn't consistent, and hence the relations I have employed(regardless whether they agree with a traditional operator or not) hold true in their given respects while our notion of sum has not varied(except for the leap it demands such that it be understood in a proper sense), further this sense, hence definition is all important as to avoid misunderstanding. My 'sum' may, given my connotation be the constant of the Euler MacLaurin formula (e.g. for [tex]{\zeta}(1)[/tex]) or an extended limit(e.g. in Euler-Abel summation)
 
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If anyone's interested, I'm taking a look at divergent series and resummations at http://mathrants.blogspot.com [Broken]
 
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