Dirichlet characters and L-functions

Dirichlet characters, conductors, primitive and quadratic characters, Dirichlet series continuation, L-function bounds, Artin and Selberg-class functions.

• a-points of Selberg class functions
• Analytic continuation of Dirichlet series
• Artin L-functions
• Conductors and primitive Dirichlet characters
• Dedekind zeta function and polar density
• Moments of Dedekind zeta coefficients over sums of six squares
• Perron's formula
• Subconvexity bounds for Dirichlet L-functions

TheoremFunctional equation of the Dirichlet beta function#✓fully proved

Let \(\chi\) be the non-principal Dirichlet character modulo \(4\), so \(\chi(1) = 1\), \(\chi(3) = -1\) and \(\chi(0) = \chi(2) = 0\); let \(\beta(s) = L(s, \chi)\) be its Dirichlet \(L\)-function (the Dirichlet beta function, continued to all of \(\mathbb{C}\)); and put \(E^*(s) = 2\,\Gamma(s)\,\beta(s) / (\pi/2)^s\). Then the functions \(\cos(\pi r/2)\, E^*(1 - r)\), \((\pi/2)^r E^*(r) / \Gamma(r)\) and \(\cos(\pi r/2)\, (\pi/2)^{r-1}\, \Gamma(1 - r)\, \beta(1 - r)\) of \(r \in \mathbb{C}\) are meromorphic on \(\mathbb{C}\), and outside a closed discrete subset of \(\mathbb{C}\) we have \(\cos(\pi r/2)\, E^*(1 - r) = 2\beta(r) = (\pi/2)^r E^*(r) / \Gamma(r)\) and \(\beta(r) = \cos(\pi r/2)\, (\pi/2)^{r-1}\, \Gamma(1 - r)\, \beta(1 - r)\).

theorem ramanujan_part1_ch7_entry18_bernoullidiff :
    MeromorphicOn chapter7Entry18Left Set.univ ∧
      MeromorphicOn chapter7Entry18Right Set.univ ∧
      MeromorphicOn chapter7Entry18FunctionalRight Set.univ ∧
      chapter7Entry18Left =ᶠ[codiscrete ℂ] chapter7Entry18Middle ∧
      chapter7Entry18Middle =ᶠ[codiscrete ℂ] chapter7Entry18Right ∧
      chapter7DirichletBeta =ᶠ[codiscrete ℂ]
        chapter7Entry18FunctionalRight

Sources

• Bruce C. Berndt, Ramanujan's Notebooks, Part I, Chapter 7.

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