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A zero density result for the Riemann zeta function

Habiba Kadiri (2013)

Acta Arithmetica

We prove an explicit bound for N(σ,T), the number of zeros of the Riemann zeta function satisfying ℜ𝔢 s ≥ σ and 0 ≤ ℑ𝔪 s ≤ T. This result provides a significant improvement to Rosser's bound for N(T) when used for estimating prime counting functions.

CM liftings of supersingular elliptic curves

Ben Kane (2009)

Journal de Théorie des Nombres de Bordeaux

Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by 𝒪 D to supersingular elliptic curves in characteristic p is surjective. In the algorithm we first determine an explicit constant D p so that | D | > D p implies that the map is necessarily surjective and then we compute explicitly the cases | D | < D p .

Computations of Galois representations associated to modular forms of level one

Peng Tian (2014)

Acta Arithmetica

We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12,20,22. All the results are rigorously proved to be correct. As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved uper bound on...

Computing fundamental domains for Fuchsian groups

John Voight (2009)

Journal de Théorie des Nombres de Bordeaux

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ , including an explicit finite presentation for Γ .

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

David J. Platt, Sumaia Saad Eddin (2013)

Colloquium Mathematicae

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.

Fonction sommatoire de la fonction de Möbius, 3. Majorations asymptotiques effectives fortes

M. El Marraki (1995)

Journal de théorie des nombres de Bordeaux

On établit les majorations M ( x ) 0 . 002969 x ( log x ) 1 / 2 , valable pour x 142194 , M ( x ) 0 . 6437752 x log x qui est la meilleure majoration possible en x log x valable pour tout x > 1 ( M ( 5 ) = 2 = 0 . 6437752 × 5 log 5 ) , et d’autres analogues. On montre enfin comment trouver des majorations effectives M ( x ) > c k x ( log log x ) 2 k ( log x ) k pour tout k .

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