On the values of the divisor function

Luca, Florian; Shparlinski, Igor E.

doi:10.1007/s00605-007-0511-3

Cited by 5 publications

(2 citation statements)

References 8 publications

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“…Except for the constant factor, τ is asymptotically not more than this value (Hardy & Wright, 1985, Theorem 317). Luca & Shparlinski (2008) give general results on the possible values of τ . It follows that #C…”

Section: Let C (S)mentioning

confidence: 99%

Compositions and collisions at degreep2

Blankertz¹,

Gathen²,

Ziegler³

2013

Journal of Symbolic Computation

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A univariate polynomial f over a field is decomposable if f = g • h = g(h) for nonlinear polynomials g and h. In order to count the decomposables, one wants to know, under a suitable normalization, the number of equal-degree collisions of the formlisions only occur in the wild case, where the field characteristic p divides deg f . Reasonable bounds on the number of decomposables over a finite field are known, but they are less sharp in the wild case, in particular for degree p 2 . We provide a classification of all polynomials of degree p 2 with a collision. It yields the exact number of decomposable polynomials of degree p 2 over a finite field of characteristic p. We also present an efficient algorithm that determines whether a given polynomial of degree p 2 has a collision or not.

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Section: Let C (S)mentioning

confidence: 99%

Compositions and collisions at degreep2

Blankertz¹,

Gathen²,

Ziegler³

2013

Journal of Symbolic Computation

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“…where e m (z) = e 2πiz/m and τ (n) = d|n 1 counts the number of divisors of n. Arithmetic properties of the divisor function have been considered in a number of works, see for example [5,6,9,13], although we are concerned mainly with congruence properties of the divisor function, which have also been considered in [4,15,16]. Exponential sums over some other arithmetic functions have been considered in [1,2].…”

Section: Introductionmentioning

confidence: 99%

Rational exponential sums over the divisor function

Kerr¹

2013

Preprint

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We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function τ (n), counting the number of divisors of n. This is done using some ideas of Sathe concerning the distribution in residue classes of the function ω(n), counting the number of prime factors of n, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.

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On Karatsuba’s problem concerning the divisor function

Korolev

2011

Monatsh Math

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On the values of the divisor function

Cited by 5 publications

References 8 publications

Compositions and collisions at degreep2

Compositions and collisions at degreep2

Rational exponential sums over the divisor function

On Karatsuba’s problem concerning the divisor function

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