Eistenstein series E_k(t=0) quick q? Modular forms

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SUMMARY

The Eisenstein series \( E_k(t) \) is defined as \( E_k(t) = 1 - \frac{2k}{B_k} \sum_{n=1}^{\infty} \sigma_{k-1}(n) q^n \), where \( B_k \) represents the Bernoulli number and \( q^n = e^{2 \pi i n t} \). At \( t=0 \), the series simplifies to \( E_k(0) = 1 \). The limit as \( t \) approaches \( i\infty \) results in \( \lim_{t \to i\infty} E_k(t) = 0 \), confirming that \( \lim_{q \to 0} E_k(t) = 0 \). This discussion focuses on the properties and implications of the Eisenstein series in the context of modular forms.

PREREQUISITES
  • Understanding of modular forms
  • Familiarity with Bernoulli numbers
  • Knowledge of the properties of the sigma function \( \sigma_{k-1}(n) \)
  • Basic concepts of complex analysis, particularly limits involving complex variables
NEXT STEPS
  • Study the derivation and applications of Eisenstein series in number theory
  • Explore the relationship between modular forms and elliptic curves
  • Investigate the properties of Bernoulli numbers and their significance in series expansions
  • Learn about the sigma function \( \sigma_{k-1}(n) \) and its role in partition theory
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Mathematicians, number theorists, and students studying modular forms and their applications in advanced mathematics.

binbagsss
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I have in my lecture notes that ##E_{k}(t=0) =1 ##,
##E_k (t)## the Eisenstein series given by:

##E_k (t) = 1 - \frac{2k}{B_k} \sum\limits^{\infty}_1 \sigma_{k-1}(n) q^{n} ##

##B_k## Bermouli number

##q^n = e^{ 2 \pi i n t} ##

context modular forms. Also have set ##lim t \to i\infty = 0## , i.e ##lim q \to 0 = 0##
##n=0## sets this to ##1##

so I have

##E_k (t) = 1 - \frac{2k}{B_k} \sum\limits^{\infty}_1 \sigma_{k-1}(n) ## ??
 
bump. thank you .
 

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