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1997
Let x denote a Dirichlet character and L(s, x) the associated Dirichlet L-function. Let us begin by considering how one would approach the problem of showing that L(1/2, x) ≠ 0. In the following, we assume that x is defined modulo a prime q. We first study the average $$ \sum\limits_{{X\left( {\bmod q} \right)}} {L\left( {\frac{1}{2},x} \right).} $$
M. Ram Murty, V. Kumar Murty
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Let x denote a Dirichlet character and L(s, x) the associated Dirichlet L-function. Let us begin by considering how one would approach the problem of showing that L(1/2, x) ≠ 0. In the following, we assume that x is defined modulo a prime q. We first study the average $$ \sum\limits_{{X\left( {\bmod q} \right)}} {L\left( {\frac{1}{2},x} \right).} $$
M. Ram Murty, V. Kumar Murty
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1997
In this section, we shall collect together a few group theoretic preliminaries. We begin by reviewing the basic aspects of characters and class functions.
M. Ram Murty, V. Kumar Murty
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In this section, we shall collect together a few group theoretic preliminaries. We begin by reviewing the basic aspects of characters and class functions.
M. Ram Murty, V. Kumar Murty
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1982
The theory of analytic functions has many applications in number theory. A particularly spectacular application was discovered by Dirichlet who proved in 1837 that there are infinitely many primes in any arithmetic progression b, b + m, b + 2m, … , where (m, b) = 1. To do this he introduced the L-functions which bear his name.
Kenneth Ireland, Michael Rosen
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The theory of analytic functions has many applications in number theory. A particularly spectacular application was discovered by Dirichlet who proved in 1837 that there are infinitely many primes in any arithmetic progression b, b + m, b + 2m, … , where (m, b) = 1. To do this he introduced the L-functions which bear his name.
Kenneth Ireland, Michael Rosen
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2013
In this chapter the adelic interpretation of modular forms is used to give an adelic description of their L-functions, which, as a byproduct, are vastly generalized. These general automorphic L-functions are defined as Euler products, the factor at a prime p is obtained from a p-adic representation, using the so called Satake Transformation.
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In this chapter the adelic interpretation of modular forms is used to give an adelic description of their L-functions, which, as a byproduct, are vastly generalized. These general automorphic L-functions are defined as Euler products, the factor at a prime p is obtained from a p-adic representation, using the so called Satake Transformation.
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Bulletin of the London Mathematical Society, 1993
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Building on the geometric--analytic--information foundation of S2--S6, we establish an interface template for general $L$-functions (including Dirichlet, elliptic curves, etc.): organized around ``Euler product--completed function--functional equation'', we provide categorical bookkeeping for degree and conductor; under S3's $\Gamma/\pi$ normalization,
Ma, Haobo, Zhang, Wenlin
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Ma, Haobo, Zhang, Wenlin
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2004
Let χ : (ℤ/mℤ)× → ℂ× be a primitive Dirichlet character modulo m. Let K = ℚ(ζ), where ζ = e 2πi/m . The identification G = Gal(K/ℚ) ≃ (ℤ/mℤ)× allows us to attach to χ a character χ Gal : G → ℂ× satisfying 1.1 if (p, m) = 1 and σ p is the Frobenius automorphism at p (the canonical generator of the decomposition group of p in G, which induces on the ...
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Let χ : (ℤ/mℤ)× → ℂ× be a primitive Dirichlet character modulo m. Let K = ℚ(ζ), where ζ = e 2πi/m . The identification G = Gal(K/ℚ) ≃ (ℤ/mℤ)× allows us to attach to χ a character χ Gal : G → ℂ× satisfying 1.1 if (p, m) = 1 and σ p is the Frobenius automorphism at p (the canonical generator of the decomposition group of p in G, which induces on the ...
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