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In mathematics, a Dirichlet ''L''-series is a function of the form : Here χ is a Dirichlet character and ''s'' a complex variable with real part greater than 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet ''L''-function and also denoted ''L''(''s'', χ). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that is non-zero at ''s'' = 1. Moreover, if χ is principal, then the corresponding Dirichlet ''L''-function has a simple pole at ''s'' = 1. == Zeros of the Dirichlet L-functions == If χ is a primitive character with χ(−1) = 1, then the only zeros of ''L''(''s'',χ) with Re(''s'') < 0 are at the negative even integers. If χ is a primitive character with χ(−1) = −1, then the only zeros of ''L''(''s'',χ) with Re(''s'') < 0 are at the negative odd integers. Up to the possible existence of a Siegel zero, zero-free regions including and beyond the line Re(''s'') = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet ''L''-functions: for example, for χ a non-real character of modulus ''q'', we have : for β + iγ a non-real zero. Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis, so the Dirichlet ''L''-functions are conjectured to obey the generalized Riemann hypothesis. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dirichlet L-function」の詳細全文を読む スポンサード リンク
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