Displaying similar documents to “On the mean square of the divisor function in short intervals”

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...

On the counting function for the generalized Niven numbers

Ryan Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño (2009)

Journal de Théorie des Nombres de Bordeaux

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Given an integer base q ≥ 2 and a completely q -additive arithmetic function f taking integer values, we deduce an asymptotic expression for the counting function N f ( x ) = # 0 ≤ n < x | f ( n ) ∣ n under a mild restriction on the values of f . When f = s q , the base q sum of digits function, the integers counted by N f are the so-called base q Niven numbers, and our result provides a generalization of the asymptotic known in that case.

Generators for the elliptic curve y 2 = x 3 - n x

Yasutsugu Fujita, Nobuhiro Terai (2011)

Journal de Théorie des Nombres de Bordeaux

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Let E be an elliptic curve given by y 2 = x 3 - n x with a positive integer n . Duquesne in 2007 showed that if n = ( 2 k 2 - 2 k + 1 ) ( 18 k 2 + 30 k + 17 ) is square-free with an integer k , then certain two rational points of infinite order can always be in a system of generators for the Mordell-Weil group of E . In this paper, we generalize this result and show that the same is true for infinitely many binary forms n = n ( k , l ) in ℤ [ k , l ] .

A note on the diophantine equation k 2 - 1 = q n + 1

Maohua Le (1998)

Colloquium Mathematicae

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In this note we prove that the equation k 2 - 1 = q n + 1 , q ≥ 2 , n ≥ 3 , has only finitely many positive integer solutions ( k , q , n ) . Moreover, all solutions ( k , q , n ) satisfy k 10 10 182 , q 10 10 165 and n 2 · 10 17 .

The divisor problem for binary cubic forms

Tim Browning (2011)

Journal de Théorie des Nombres de Bordeaux

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We investigate the average order of the divisor function at values of binary cubic forms that are reducible over ℚ and discuss some applications.