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In this setting, plays the role of the real number line in the classical version of Poisson summation, and plays the role of the integers that appear in the sum. The generalised version of Poisson summation is called the Selberg Trace Formula, and has played a role in proving many cases of Artin's conjecture and in Wiles's proof of Fermat's ...
Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When Γ is the cocompact subgroup Z of the real numbers G = R, the Selberg trace formula is essentially the Poisson summation formula.
A version of the arithmetic Riemann–Roch theorem states that if k is a global field, and f is a suitably admissible function of the adeles of k, then for every idele a, one has a Poisson summation formula:
Then, the Poisson summation formula for f is used to compare the density of the optimal lattice with that of any other packing. [16] Before the proof had been formally refereed and published, mathematician Peter Sarnak called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct." [17]
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X 1. [citation needed] Every infinitely divisible probability distribution is a limit of compound Poisson distributions. [1] And compound Poisson distributions is infinitely divisible by the ...
Applying the Poisson summation formula to the characteristic function of the polygon leads to another proof. [16] Pick's theorem was included in a 1999 web listing of the "top 100 mathematical theorems", which later became used by Freek Wiedijk as a benchmark set to test the power of different proof assistants.
In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups .