Displaying similar documents to “Landau’s function for one million billions”

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 [ k , l ] .

Approximation of values of hypergeometric functions by restricted rationals

Carsten Elsner, Takao Komatsu, Iekata Shiokawa (2007)

Journal de Théorie des Nombres de Bordeaux

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We compute upper and lower bounds for the approximation of hyperbolic functions at points 1 / s ( s = 1 , 2 , ) by rationals x / y , such that x , y satisfy a quadratic equation. For instance, all positive integers x , y with y 0 ( mod 2 ) solving the Pythagorean equation x 2 + y 2 = z 2 satisfy | y sinh ( 1 / s ) - x | log log y log y . Conversely, for every s = 1 , 2 , there are infinitely many coprime integers x , y , such that | y sinh ( 1 / s ) - x | log log y log y and x 2 + y 2 = z 2 hold simultaneously for some integer z . A generalization to the approximation of h ( e 1 / s ) for rational...

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 and discuss some applications.

On the mean square of the divisor function in short intervals

Aleksandar Ivić (2009)

Journal de Théorie des Nombres de Bordeaux

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We provide upper bounds for the mean square integral X 2 X 𝔻 k ( x + h ) - 𝔻 k ( x ) 2 d x , where h = h ( X ) 1 , h = o ( x ) as X and h lies in a suitable range. For k 2 a fixed integer, 𝔻 k ( x ) is the error term in the asymptotic formula for the summatory function of the divisor function d k ( n ) , generated by ζ k ( s ) .