Geometric properties of points on modular hyperbolas
Authors:
Kevin Ford, Mizan R. Khan and Igor E. Shparlinski
Journal:
Proc. Amer. Math. Soc. 138 (2010), 41774185
MSC (2010):
Primary 11A07; Secondary 11H06, 11N69
Published electronically:
July 9, 2010
MathSciNet review:
2680044
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Given an integer , let be the set and let be the maximal difference of for . We prove that for almost all , We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of .
 1.
W.
R. Alford, Andrew
Granville, and Carl
Pomerance, There are infinitely many Carmichael numbers, Ann.
of Math. (2) 139 (1994), no. 3, 703–722. MR 1283874
(95k:11114), http://dx.doi.org/10.2307/2118576
 2.
George
E. Andrews, A lower bound for the volume of
strictly convex bodies with many boundary lattice points, Trans. Amer. Math. Soc. 106 (1963), 270–279. MR 0143105
(26 #670), http://dx.doi.org/10.1090/S00029947196301431057
 3.
Kevin
Ford, The distribution of integers with a divisor in a given
interval, Ann. of Math. (2) 168 (2008), no. 2,
367–433. MR 2434882
(2009m:11152), http://dx.doi.org/10.4007/annals.2008.168.367
 4.
Kevin
Ford, Mizan
R. Khan, Igor
E. Shparlinski, and Christian
L. Yankov, On the maximal difference between an
element and its inverse in residue rings, Proc.
Amer. Math. Soc. 133 (2005), no. 12, 3463–3468. MR 2163580
(2006c:11108), http://dx.doi.org/10.1090/S0002993905079621
 5.
Étienne
Fouvry, Sur le problème des diviseurs de Titchmarsh, J.
Reine Angew. Math. 357 (1985), 51–76 (French). MR 783533
(87b:11090), http://dx.doi.org/10.1515/crll.1985.357.51
 6.
Glyn
Harman, On the number of Carmichael numbers up to 𝑥,
Bull. London Math. Soc. 37 (2005), no. 5,
641–650. MR 2164825
(2006d:11106), http://dx.doi.org/10.1112/S0024609305004686
 7.
D.
R. HeathBrown, Zerofree regions for Dirichlet
𝐿functions, and the least prime in an arithmetic progression,
Proc. London Math. Soc. (3) 64 (1992), no. 2,
265–338. MR 1143227
(93a:11075), http://dx.doi.org/10.1112/plms/s364.2.265
 8.
Mizan
R. Khan and Igor
E. Shparlinski, On the maximal difference between an element and
its inverse modulo 𝑛, Period. Math. Hungar.
47 (2003), no. 12, 111–117. MR 2024977
(2004k:11006), http://dx.doi.org/10.1023/B:MAHU.0000010815.14847.96
 9.
Mizan
R. Khan, Igor
E. Shparlinski, and Christian
L. Yankov, On the convex closure of the graph of modular
inversions, Experiment. Math. 17 (2008), no. 1,
91–104. MR
2410119 (2009e:11003)
 10.
Igor
E. Shparlinski, On the distribution of points on multidimensional
modular hyperbolas, Proc. Japan Acad. Ser. A Math. Sci.
83 (2007), no. 2, 5–9. MR 2303621
(2008g:11161)
 11.
Igor
E. Shparlinski, Distribution of modular inverses and multiples of
small integers and the SatoTate conjecture on average, Michigan Math.
J. 56 (2008), no. 1, 99–111. MR 2433659
(2009e:11154), http://dx.doi.org/10.1307/mmj/1213972400
 12.
Terence
Tao, Structure and randomness, American Mathematical Society,
Providence, RI, 2008. Pages from year one of a mathematical blog. MR 2459552
(2010h:00002)
 1.
 W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Annals of Math. (2) 139 (1994), 703722. MR 1283874 (95k:11114)
 2.
 G. Andrews, A lower bound for the volume of strictly convex bodies with many boundary lattice points, Trans. Amer. Math. Soc. 106 (1963), 270279. MR 0143105 (26:670)
 3.
 K. Ford, The distribution of integers with a divisor in a given interval, Annals. of Math. (2) 168 (2008), 367433. MR 2434882 (2009m:11152)
 4.
 K. Ford, M. R. Khan, I. E. Shparlinski and C. L. Yankov, On the maximal difference between an element and its inverse in residue rings, Proc. Amer. Math. Soc. 133 (2005), 34633468. MR 2163580 (2006c:11108)
 5.
 É. Fouvry, Sur le problème des diviseurs de Titchmarsh, J. Reine Angew. Math. 357 (1985), 5176. MR 783533 (87b:11090)
 6.
 G. Harman, On the number of Carmichael numbers up to , Bull. London Math. Soc. 37 (2005), 641650. MR 2164825 (2006d:11106)
 7.
 D. R. HeathBrown, Zerofree regions for Dirichlet functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64 (1992) 265338. MR 1143227 (93a:11075)
 8.
 M. R. Khan and I. E. Shparlinski, On the maximal difference between an element and its inverse modulo , Periodica Math. Hungarica 47 (2003), 111117. MR 2024977 (2004k:11006)
 9.
 M. R. Khan, I. E. Shparlinski and C. L. Yankov, On the convex closure of the graph of modular inversions, Experimental Math. 17 (2008), 91104. MR 2410119 (2009e:11003)
 10.
 I. E. Shparlinski, On the distribution of points on multidimensional modular hyperbolas, Proc. Japan Acad. Sci., Ser. A 83 (2007), 59. MR 2303621 (2008g:11161)
 11.
 I. E. Shparlinski, Distribution of modular inverses and multiples of small integers and the SatoTate conjecture on average, Michigan Math. J. 56 (2008), 99111. MR 2433659 (2009e:11154)
 12.
 T. Tao, Structure and Randomness, Amer. Math. Soc., Providence, RI, 2008. MR 2459552
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
11A07,
11H06,
11N69
Retrieve articles in all journals
with MSC (2010):
11A07,
11H06,
11N69
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
DOI:
http://dx.doi.org/10.1090/S000299392010105610
PII:
S 00029939(2010)105610
Received by editor(s):
February 11, 2010
Published electronically:
July 9, 2010
Additional Notes:
The research of the first author was supported in part by NSF grants DMS0555367 and DMS0901339.
The research of the third author was supported by ARC grants DP0556431 and DP1092835.
Communicated by:
Matthew A. Papanikolas
Article copyright:
© Copyright 2010
American Mathematical Society
