Displaying similar documents to “Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis”

Dihedral and cyclic extensions with large class numbers

Peter J. Cho, Henry H. Kim (2012)

Journal de Théorie des Nombres de Bordeaux

Similarity:

This paper is a continuation of []. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups D n , n = 3 , 4 , 5 , and cyclic groups C n , n = 4 , 5 , 6 . We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding L -functions...

Ternary quadratic forms with rational zeros

John Friedlander, Henryk Iwaniec (2010)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We consider the Legendre quadratic forms ϕ a b ( x , y , z ) = a x 2 + b y 2 - z 2 and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers 1 ≤ a ≤ A , 1 ≤ b ≤ B , for which the form ϕ a b has a non-trivial rational zero. Under certain mild conditions on the integers a , b , we are able to find the asymptotic formula for the number of such forms.

A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip 3 / 4 ≤ σ < 1

Yuk-Kam Lau (2006)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let E σ ( T ) be the error term in the mean square formula of the Riemann zeta-function in the critical strip 1 / 2 < σ < 1 . It is an analogue of the classical error term E ( T ) . The research of E ( T ) has a long history but the investigation of E σ ( T ) is quite new. In particular there is only a few information known about E σ ( T ) for 3 / 4 < σ < 1 . As an exploration, we study its mean value ∫ 1 T E σ ( u ) d u . In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of T .

Landau’s function for one million billions

Marc Deléglise, Jean-Louis Nicolas, Paul Zimmermann (2008)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let 𝔖 n denote the symmetric group with n letters, and g ( n ) the maximal order of an element of 𝔖 n . If the standard factorization of M into primes is M = q 1 α 1 q 2 α 2 ... q k α k , we define ℓ ( M ) to be q 1 α 1 + q 2 α 2 + ... + q k α k ; one century ago, E. Landau proved that g ( n ) = max ℓ ( M ) ≤ n M and that, when n goes to infinity, log g ( n ) ∼ n log ( n ) . There exists a basic algorithm to compute g ( n ) for 1 ≤ n ≤ N ; its running time is 𝒪 N 3 / 2 / log N and the needed memory is 𝒪 ( N ) ; it allows computing g ( n ) up to, say, one million. We describe an algorithm to calculate g ( n ) for n up to 10 15 . The main idea is to use the...