bio  website  math.umn.edu/~garrett 

location  US  
age  63  
visits  member for  4 years, 3 months 
seen  13 mins ago  
stats  profile views  5,402 
I am interested in the application of modern analysis to the spectral theory of Eisenstein series and other automorphic forms, Lfunctions, and number theory. This includes representation theory of (mostly reductive) Lie groups and padic groups, and harmonic analysis on these groups and their homogeneous spaces. Necessarily Schwartz' distributions, Sobolev spaces, and similar things play important supporting roles. I find that a mildly categorical viewpoint is very helpful in keeping things straight.
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ZetaDeterminant for shifted Laplacians on the circle
Repaired the computational errors since the question got bumped upward anyway... 
21h

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ZetaDeterminant for shifted Laplacians on the circle
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ZetaDeterminant for shifted Laplacians on the circle
Some minor computational booboos in the above: should be $e^{\pi(n^2+c)y}$, of course, and pursuant... doesn't change it qualitatively. 
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Is the twisted symmetric fifth power $L$function holomorphic?
Following up on @Marty's comment: in the 80s, several people, including PiateskiShapiroRallis, myself, and others, tried to identify what sort of RankinSelberg situation could produce such "higher" Lfunctions... tentatively thinking in terms of "generalized groups"... but/and found that (at least it seemed at the time) there was no sane "generalized group" recipe that could produce a given Lfunction "at will". Maybe there has been progress, but it is already not so easy to make a reductive "group" with arbitrarily specified Coxeter group as "Weyl group"... Such obstacles. 
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ZetaDeterminant for shifted Laplacians on the circle
Also, an EulerMacLaurin summation approach might work as well as anything, depending, ... 
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answered  ZetaDeterminant for shifted Laplacians on the circle 
Jul 27 
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ZetaDeterminant for shifted Laplacians on the circle
It seems to me that a variant of Riemann's integral representation of $\zeta$ using $\theta$, the latter's functional equation from Poisson summation, gives the meromorphic continuation... but, unlike zeta itself, superficially it appears difficult to get information about the derivative at $0$, but is that what's really desired, or is it more the residue at the pole at $s=1$? Can you clarify "in terms of what" you'd really like the outcome? 
Jul 22 
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Analytic Number Theory without Pigeonhole Principle
It is still not clear to me what you mean by "analytic number theory". Please clarify? There is much potential ambiguity in that label. E.g., do you mean IwasawaTate theory of zeta functions? JacquetLanglands? Sieves? The ZhangMaynardTao business? Subtle things about moment estimates? Subconvexity? "The thing is", most of these are not really "combinatorial" or "discrete" in any operational sense, so there's some element of misreference, to my mind. Please clarify? 
Jul 22 
answered  Analytic Number Theory without Pigeonhole Principle 
Jul 21 
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Make mathematical sense of the Dirac well Potential Equation
@ChristianRemling, yes, indeed, thanks... and to give an exampleobjection: one way to interpret the (mathematical) difficulty in interpreting $\delta'$ as a "potential" in analogous fashion is that it is not inside $H^{1}$, and it itself cannot be "legally" applied to solutions to $(\Delta\lambda)u=\delta'$, since these solutions will only be in $H^{1/2\epsilon}$, not $H^{3/2+\epsilon}$, etc. 
Jul 21 
revised 
Make mathematical sense of the Dirac well Potential Equation
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Jul 21 
revised 
Make mathematical sense of the Dirac well Potential Equation
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Jul 21 
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Make mathematical sense of the Dirac well Potential Equation
@JeanDuchon, yes, what I wrote was careless. But/and I think the issue is not truly about literal pointwise multiplication, for a reason I will try to clarify in the edit I will do just now... 
Jul 20 
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Make mathematical sense of the Dirac well Potential Equation
Exactly. Dirac's marvelous intuition in the late 1920s was nicely rigorized in one fashion by (B.Levi) Sobolev's spaces by the 1930s, as promoted and amplified by Gelfandetal in the "Generalized Functions" 6 volumes. 
Jul 18 
awarded  modularforms 
Jul 17 
answered  How to construct the symmetric power function from a modular form? 
Jul 17 
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How to construct the symmetric power function from a modular form?
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Jul 16 
revised 
Make mathematical sense of the Dirac well Potential Equation
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Jul 16 
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Make mathematical sense of the Dirac well Potential Equation
Ooops, yes, I'll change that sign on the epsilon. The general, basic things about Sobolev spaces are treated in most or many books on PDEs, especially linear ones. Folland's book (or Tata lectures), Brezis' book, are two that do this sort of thing. The question of multiplication is an immediate corollary, then, since $H^{+s}$ and $H^{s}$ (in various contexts) are in duality, so "pair" to $L^1$, at least. The Sobolev imbedding business is treated in those sources, certainly. Googling "Sobolev space" should give lots of useful results, too. 
Jul 16 
answered  Make mathematical sense of the Dirac well Potential Equation 