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Elliptic functions proof -- convergence series on lattice

  1. Feb 19, 2017 #1
    1. The problem statement, all variables and given/known data
    Hi

    I am looking at the proof attached for the theorem attached that:

    If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2##
    where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##.

    For any integer ##r \geq 0 ## :

    ##\Omega_r := {mw_1+nw_2|m,n \in Z, max {|m|,|n|}=r} ##
    ##\Pi_r := {mw_1+nw_2|m,n \in Z, max {|m|,|n|}=r} ##

    so that ##\Omega = {0} \Cup \Omega_1 \Cup \Omega_2 \Cup....##

    Each ##\Omega_r## has cardinality ##8r##

    QUESTIONS
    - To prove via the comparison test, we only need to bound from above by a series that converges, so why have we bound from above and below - this is my main question really, why have we bound from above and below
    - Does this proove via both the convergence test and the Weierstass-M test? Since each term in the sequence ##|w|^{-s}## is bound above by a real constant.
    - The definition of the W-M test is ##u_n## a seqence of functions, if for each a ##n \in N## there exists ##M_n \in R## satisfying ##|u_n(z)|\leq M_n ## got all ##z \in E## where ##u_n : E \to C## and ##\sum M_n## converges. Here the '##u_n##' are ##|w|## are already taken the absolute value, does this change anything here or the W-M test or does it still apply in the same way?

    Many thanks in advance.


    2. Relevant equations
    as above

    3. The attempt at a solution
    as above
     

    Attached Files:

  2. jcsd
  3. Feb 22, 2017 #2
    bump.many thanks.
     
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