(August 29, 2013)
Counting zeros of (s)
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/counting zeros of zeta.pdf]
Zeros of (s) in the critical strip 0 Re (s) 1 ar
(May 5, 2011)
Continuous spectrum for SL2(Z)\H
Paul Garrett [email protected]
1.
2.
3.
4.
5.
6.
http:/
/www.math.umn.edu/ garrett/
Pseudo-Eisenstein series adjunction to constant term
Decomposition of pseudo-Eisenstein series: beginning
Recollection of
(April 7, 2011)
Modular forms and number theory exercises 16
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[mfms 16.1] Some calculus: derive the ordinary dierential equation
f (|x|) = f (|x|) +
n1
f (|x|)
|x|
[mfms 16.2] More calcul
(January 7, 2012)
Transition exercise on Eisenstein series
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
1. Rewriting the GL2 Eisenstein series
2. Application: constant term of GL2 Eisenstein series
3. Application: Hecke operators o
(June 7, 2011)
The simplest Eisenstein series
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
1. Statements of results
2. Proofs
We explain some essential aspects of the simplest Eisenstein series for SL2 (Z) on the upper half-plane H
(January 6, 2011)
(1 q n)24
The product expansion (z) = q
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
1. Weils proof
2. Appendix: Siegels proof
[Weil 1968] used a converse theorem to reprove the product expansion for the unique ho
(June 6, 2011)
Introduction to zeta integrals and L-functions for GLn
Paul Garrett [email protected]
1.
2.
3.
4.
http:/
/www.math.umn.edu/ garrett/
Fourier-Whittaker expansions of cuspforms on GLr
The Hecke-type case GLn GLn1
The Rankin-Selberg case GL
(April 16, 2011)
Spheres and hyperbolic spaces
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
Basic examples of non-Euclidean geometries are best studied by studying the groups that preserve the
geometries. In fact, rather than speci
(August 24, 2013)
01b. Product expansion of sin x
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/01b product expn sine.pdf]
Euler did eventually prove the product
(August 30, 2013)
Asymptotics of integrals
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is
http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/02d asymptotics of integrals.pdf]
Heuristic for Stirlings asymptotic
Wa
(August 30, 2013)
Phragmn-Lindelf Theorems
e
o
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is
http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/02e Phragmen-Lindelof.pdf]
E. Phragmn, E. Lindelf, Sur une extension
(August 24, 2013)
Euler product of (s)
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/01a Euler product.pdf]
Eulers discovery that
1
=
ns
n=1
p prime
1
1 ps
(for R
(August 22, 2013)
01. Euler and (s)
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/01 Euler and zeta.pdf]
[0.1] Summing series
Innite sums that telescope can be un
(September 21, 2013)
The keyhole/Hankel contour and (n)
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/02g keyhole and zeta.pdf]
The contour-integration trick illu
(September 12, 2013)
The Estermann phenomenon
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
[This document is
http:/www.math.umn.edu/garrett/m/mfms/notes 2013-14/03f estermann phenom.pdf]
The Estermann phenomenon is that not every n
(June 6, 2011)
Meromorphic continuation and functional equation of GL2 Eisenstein series
Paul Garrett [email protected]
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/www.math.umn.edu/ garrett/
Poisson summation suces to give a general argument for meromorphic continuation and functional eq