| 1. |
- Kurlberg, Pär, et al.
(author)
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On a problem of Arnold : The average multiplicative order of a given integer
- 2013
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In: Algebra & Number Theory. - : Mathematical Sciences Publishers. - 1937-0652 .- 1944-7833. ; 7:4, s. 981-999
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Journal article (peer-reviewed)abstract
- For coprime integers g and n, let l(g) (n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l(g) (n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l(g) (p) as p <= x ranges over primes.
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| 2. |
- Kurlberg, Par, et al.
(author)
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On Sets of Integers Which Are Both Sum-Free and Product-Free
- 2013
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In: Integers. - 1553-1732.
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Journal article (peer-reviewed)abstract
- We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
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| 3. |
- Kurlberg, Pär, et al.
(author)
-
The Maximal Density of Product-Free Sets in Z/nZ
- 2013
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In: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; :4, s. 827-845
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Journal article (peer-reviewed)abstract
- This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab=c (mod n), with a, b, c being in the set. In a previous paper, we constructed an infinite sequence of integers (n(i))(i >= 1) and product-free sets S-i in Z/n(i)Z such that the density vertical bar S-i vertical bar/n(i) -> 1 as i -> infinity, where vertical bar S-i vertical bar denotes the cardinality of S-i. Here, we obtain matching, up to constants, upper and lower bounds on the maximal attainable density as n -> infinity.
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