Displaying similar documents to “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 ”

Ternary quadratic forms with rational zeros

John Friedlander, Henryk Iwaniec (2010)

Journal de Théorie des Nombres de Bordeaux

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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.

Landau’s function for one million billions

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

Journal de Théorie des Nombres de Bordeaux

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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...