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Are there any cases when Abel, Cesaro, Borel, Ramanujan, Zeta regularizations are applicable for regularization of a divergent series or integral but give different results?

Anixx
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    If there were such a case it should be mentioned in wikipedia or mathworld or any serious online/paper encyclopedia... So I'm sure, there is no such case (except, if you like, that for some regularization the "value" is infinity and for some other is a finite value (multivaluedness in the sense, that only *one* possible finite value occurs and else only infinities) – Gottfried Helms Aug 29 '19 at 19:29
  • @Gottfried Helms I need a reference for paper that these approaches are equal. – Anixx Aug 29 '19 at 19:42
  • Maybe -but not too modern- Konrad Knopp, "On infinite series" (or so) - it is online in german language, but I think I've come across the english translation as well. Chapter XIII is about divergent series. The other classic is surely G.H. Hardy's monography. But besides of discussion of the Tauberian theorems and perhaps general statements about the equivalence of methods I don't think, they've made the explicite statement in that form you wish here - I surely would remember this! (Note that very similar questions appeared here and in math-overflow from time to time, I think to remember (...) – Gottfried Helms Aug 29 '19 at 19:49
  • (...) one user who collected material with that focus for writing his batchelor-exposé.) – Gottfried Helms Aug 29 '19 at 19:50
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    Konrad Knopp, unendliche Reihen, chap XIII, pg 480: "(2) permanence-principle: a new regularization-method should first be compatible with convergent series: for convergent series it should give the same value. (2b) Usefulness: But to be useful at all, we expect that it can regularize at least one series which was not convergent itself" (3) When there are different regularization-methods which are capable to assign a finite value to a given series, then all that regularization-methods *shall give the same value* to that series. " So (3) is -so to say- an axiom which (...) – Gottfried Helms Aug 29 '19 at 20:43
  • (...) *defines* , which methods are acceptable in number-theory. Maybe this passage helps for your text (It is not translated here, I just paraphrased. If it is helpful for you you might find it in the english book likely available in google.books - I'm not so good in english to give you a reliable translation by myself) – Gottfried Helms Aug 29 '19 at 20:45
  • @Gottfried Helms does he list the methods that give the same value? – Anixx Aug 29 '19 at 20:45
  • At least Cesaro, Hölder, Abel, Euler, Riesz,Borel,Le Roy in the book. Don't know at the moment whether he touches Zeta-regularization. Ramanujan is, as far as I remember, in a follow-up article. But I think I've the german chapter locally as pdf-file. I'll see and report – Gottfried Helms Aug 29 '19 at 20:48
  • S. Chapman & G.H.Hardy (1911) is referred to an article (in "quarterly Journal, Vol 42, pg. 181" on the systematizing of the various regularization-methods. Perhaps there is an explicite statement like you need it. – Gottfried Helms Aug 29 '19 at 20:56
  • Knopp's most contribution is first part of previous century. He did not, for instance know of methods like Aitken-process etc and I do not know whether this compatibility axiom extends to those modern, non-linear, summation procedures as well. Anyway, the reference-list of Knopp's chap XIII should be understandable for you and gives a rich list of further-readings (of course of his contemporaries only...). -it's late here, I can come back to this tomorrow evening. – Gottfried Helms Aug 29 '19 at 21:11
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    It depends how you define Abel, Cesaro, Borel, Ramanujan, Zeta regularizations and how many of them do you assume well-defined for your series. So make your question precise. The first thing to know is that if $a_n= O(n^c)$ then $\sum^{Abel\ summation} a_n=A$ well-defined implies $F(s) = \sum a_n n^{-s}$ extends analytically to $\Re(s) >0$ and $\lim_{s \to 0^+} F(s)=A$. https://math.stackexchange.com/questions/3328567/divergent-sums-via-analytic-continuation-power-series-vs-dirichlet-series/3328752#3328752 @GottfriedHelms – reuns Aug 30 '19 at 10:04

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