On the maximal difference between an element and its inverse in residue rings
Authors:
Kevin Ford, Mizan R. Khan, Igor E. Shparlinski and Christian L. Yankov
Journal:
Proc. Amer. Math. Soc. 133 (2005), 34633468
MSC (2000):
Primary 11A07, 11N25
Published electronically:
June 8, 2005
MathSciNet review:
2163580
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: We investigate the distribution of where
Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on . We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study than a more traditional way using exponential sums.
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 C. Cobeli and A. Zaharescu, On the Distribution of the points on an Affine Curve in Dimensions, Acta Arithmetica 99 (2001), 321329. MR 1845688 (2003j:11066)
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 R. Hall and G. Tenenbaum, Divisors, Cambridge Tracts in Mathematics 90, Cambridge University Press, 1988. MR 0964687 (90a:11107)
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 K.H. Indlekofer and N. M. Timofeev, Divisors of Shifted Primes, Publ. Math. Debrecen 60 (2002), 307345. MR 1898566 (2003b:11076)
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Additional Information
Kevin Ford
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, 1409 West Green Street, Urbana, Illinois 61801
Email:
ford@math.uiuc.edu
Mizan R. Khan
Affiliation:
Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226
Email:
khanm@easternct.edu
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email:
igor@ics.mq.edu.au
Christian L. Yankov
Affiliation:
Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226
Email:
yankovc@easternct.edu
DOI:
http://dx.doi.org/10.1090/S0002993905079621
PII:
S 00029939(05)079621
Received by editor(s):
July 16, 2004
Published electronically:
June 8, 2005
Communicated by:
David E. Rohrlich
Article copyright:
© Copyright 2005 American Mathematical Society
