Explicit upper bounds for |L(1,χ)| for primitive even Dirichlet characters

Stéphane Louboutin

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Displaying similar documents to “Explicit upper bounds for |L(1,χ)| for primitive even Dirichlet characters”

Explicit upper bounds for |L(1,χ)| when χ(3) = 0

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Let χ be a primitive Dirichlet character of conductor q and denote by L(z,χ) the associated L-series. We provide an explicit upper bound for |L(1,χ)| when 3 divides q.

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Note on a hypothesis implying the non-vanishing of Dirichlet L-series L(s,χ) for s > 0 and real characters χ

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We prove that if χ is a real non-principal Dirichlet character for which L(1,χ) ≤ 1- log2, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that L(s,χ) > 0 for s > 0.

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Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions

Yu. Matiyasevich, F. Saidak, P. Zvengrowski (2014)

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As usual, let s = σ + it. For any fixed value of t with |t| ≥ 8 and for σ < 0, we show that |ζ(s)| is strictly decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality related to the monotonicity of all three functions is proved: ℜ (η'(s)/η(s)) < ℜ (ζ'(s)/ζ(s)) < ℜ (ξ'(s)/ξ(s)). It is also shown that extending the above monotonicity result for |ζ(s)|, |ξ(s)|, or |η(s)|...

On the Mean Value of Dirichlet Series.

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On the size of L(1,χ) and S. Chowla's hypothesis implying that L(1,χ) > 0 for s > 0 and for real characters χ

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We give explicit constants κ such that if χ is a real non-principal Dirichlet character for which L(1,χ) ≤ κ, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that L(s,χ) > 0 for s > 0. These constants are larger than the previous ones κ = 1- log 2 = 0.306... and κ = 0.367... we obtained elsewhere.

Upper bound estimate of incomplete Cochrane sum

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By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.

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On various mean values of Dirichlet L-functions

Takuya Okamoto, Tomokazu Onozuka (2015)

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We give a method of obtaining explicit formulas for various mean values of Dirichlet L-functions which are expressed in terms of the Riemann zeta-function, the Euler function and Jordan's totient functions. Applying those results to mean values of Dirichlet L-functions, we also give an explicit formula for certain mean values of double Dirichlet L-functions.