| Dirichlet L-series |
Article Index for Dirichlet |
Information AboutDirichlet L-series |
| CATEGORIES ABOUT DIRICHLET L-FUNCTION | |
| zeta and l-functions | |
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: 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). It was proven by Dirichlet that ''L''(1,χ)≠0 for all Dirichlet characters χ, allowing him to establish his Theorem On Primes In Arithmetic Progressions . 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. 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 . FUNCTIONAL EQUATION Let us assume that χ is a primitive character to the modulus ''k''. Defining : where Γ denotes the Gamma Function and the symbol ''a'' is given by : one has the Functional Equation : Here we wrote τ(χ) for the Gauss Sum : | ||
| {{cite Bookauthor | H Davenport |
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