Download PDF by Kubota T.: An elementary theory of Eisenstein series

By Kubota T.

ISBN-10: 0470509201

ISBN-13: 9780470509203

Show description

Read Online or Download An elementary theory of Eisenstein series PDF

Best number theory books

Download e-book for iPad: An Approach to the Selberg Trace Formula via the Selberg by Jürgen Fischer

The Notes supply an immediate method of the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) performing on the higher half-plane. the elemental inspiration is to compute the hint of the iterated resolvent kernel of the hyperbolic Laplacian that allows you to arrive on the logarithmic spinoff of the Selberg zeta-function.

Download PDF by Daniel Bump: Automorphic Forms and Representations

This publication covers either the classical and illustration theoretic perspectives of automorphic types in a method that's obtainable to graduate scholars getting into the sector. The remedy relies on whole proofs, which display the distinctiveness rules underlying the elemental structures. The booklet good points wide foundational fabric at the illustration conception of GL(1) and GL(2) over neighborhood fields, the speculation of automorphic representations, L-functions and complicated subject matters corresponding to the Langlands conjectures, the Weil illustration, the Rankin-Selberg process and the triple L-function, and examines this material from many various and complementary viewpoints.

Additional resources for An elementary theory of Eisenstein series

Example text

N 1/, we havef1 D 1 t t , which shows that fm is equal to the left-handP side of (8). tn (9) Writing . 1 t//m=mŠ D 1 nDm an;m nŠ , and taking the derivatives of both sides, we have . n 1/Š This shows that an;m satisfies the same recurrence formula as verify the initial conditions. n m . 7. Sometimes the Stirling numbers are defined by the formulas (2) and (3) in this proposition. 1) and (7) in this proposition, we give a proof of the formulas that were left unproved in the previous chapter (p. 17).

N 1/Š nD1 i D0 nD1 1. Thus t u which concludes the proof. 13. Conversely, if we define Bn by the formula in the above theorem, then we have ! 2). 12 are equivalent. 4. 4 revisited). If n is an odd integer greater than or equal to 3, then Bn D 0. Proof. It suffices to show that the formal power series odd-degree terms. e t 1 C 1/ D 2 et 1 t et et 1 t 2 does not have any t t t D t C 2 e 1 2 and . t/e t e t 1 . t/ t t t t D C ; C D t 2 1 et 2 e 1 2 22 t et et 1 1 Bernoulli Numbers is invariant under the substitution t !

By definition, f is a divisor of f . 3 (primitive character). A Dirichlet character primitive if f D f . 2 Generalized Bernoulli Numbers 53 Let be a Dirichlet character modulo f . Since . 1/2 D .. 1/ D 1, we see that . 1/ D ˙1. A Dirichlet character with . 1/ D 1 is called an even character. A character with . 1/ D 1 is called an odd character. a/. We have f D f . 4. Define j W Z ! j D 0; 1; 2; 3/ as follows. e. j D 0; 1; 2; 3/. j D 0; 1; 2; 3/ are all the Dirichlet characters modulo 8. D 1 2 / is not primitive, with f 3 D 4.

Download PDF sample

An elementary theory of Eisenstein series by Kubota T.

by John

Rated 4.67 of 5 – based on 34 votes