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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a problem of D. H. Lehmer


Authors: Stéphane R. Louboutin, Joël Rivat and András Sárközy
Journal: Proc. Amer. Math. Soc. 135 (2007), 969-975
MSC (2000): Primary 11K45; Secondary 11L05, 11L40
Published electronically: September 26, 2006
MathSciNet review: 2262896
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Abstract: Let $ p$ be an odd prime number. For $ n\in\{1,\ldots,p-1\}$ we denote the inverse of $ n$ modulo $ p$ by $ n^{*}$ with $ n^{*}\in\{1,\ldots,p-1\}$. Given $ \varepsilon>0$, we prove that in any range $ n\in\{N+1,\ldots,N+L\}\subseteq\{1,\ldots,p-1\}$ of length $ L\geq p^{1/2+\varepsilon}$ the probability that $ n^{*}$ has the same parity as $ n$ tends to $ 1/2$ as $ p\rightarrow +\infty$. This result was previously known only to hold true in the full range $ n\in\{1,\ldots,p-1\}$ of length $ L=p-1$. We will also obtain quantitative results on the pseudorandomness of the sequence $ (-1)^{n+n^{*}}$ for which we estimate the well-distribution $ W$ and correlation measures $ C_k$ as defined by Mauduit and Sárközy (1997).


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Additional Information

Stéphane R. Louboutin
Affiliation: Institut de Mathématiques de Luminy, UMR 6206, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email: loubouti@iml.univ-mrs.fr

Joël Rivat
Affiliation: Institut de Mathématiques de Luminy, UMR 6206, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email: rivat@iml.univ-mrs.fr

András Sárközy
Affiliation: Department of Algebra and Number Theory, Eötvös Loránd University, H-1117 Budapest, Pázmány Péter sétány 1/c, Hungary
Email: sarkozy@cs.elte.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08558-3
PII: S 0002-9939(06)08558-3
Keywords: Pseudo-random, binary sequence, correlation
Received by editor(s): August 8, 2005
Received by editor(s) in revised form: November 8, 2005
Published electronically: September 26, 2006
Additional Notes: The research of the third author was partially supported by the Hungarian National Foundation for Scientific Research, Grants No T043623 and T049693. This paper was written when the third author was visiting the Institut de Mathématiques de Luminy.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society