L-functions are complex-valued functions that encode crucial arithmetic information. They're associated with various mathematical objects, from the to Galois representations, and are defined using intricate products and determinants.
These functions have fascinating analytic properties, including convergence in specific regions and meromorphic continuation. Their zeros and poles hold key insights, with applications ranging from prime distribution to the and theorems.
L-functions and Their Analytic Properties
L-functions of Galois representations
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L-functions encode arithmetic information as complex-valued functions associated with arithmetic objects (Riemann )
Galois representations map absolute Galois group to general linear group as continuous homomorphisms over finite-dimensional vector spaces ()
L-function for a Galois representation defined as L(s,ρ)=∏p primedet(I−ρ(Frobp)p−s)−1 where s complex variable, ρ Galois representation, Frobp Frobenius element at prime p
Analytic properties of L-functions
Convergence occurs in right half-plane, represented as