On the distribution of the order over residue classes
Author:
Pieter Moree
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 121128
MSC (2000):
Primary 11N37, 11R45; Secondary 11N69
Published electronically:
August 18, 2006
MathSciNet review:
2263073
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: For a fixed rational number and integers and we consider the set of primes such that the order of modulo is congruent to . Under the Generalized Riemann Hypothesis (GRH), it can be shown that the set has a natural density . Arithmetical properties of are described, and is compared with : the average density of elements in a field of prime characteristic having order congruent to . It transpires that has a strong tendency to be equal to , or at least to be close to it.
 [CM]
Koji
Chinen and Leo
Murata, On a distribution property of the residual order of
𝑎\𝑝𝑚𝑜𝑑𝑝. I, J. Number
Theory 105 (2004), no. 1, 60–81. MR 2032442
(2005c:11117), http://dx.doi.org/10.1016/j.jnt.2003.06.005
Leo
Murata and Koji
Chinen, On a distribution property of the residual order of
𝑎\𝑝𝑚𝑜𝑑𝑝. II, J. Number
Theory 105 (2004), no. 1, 82–100. MR 2032443
(2005c:11118), http://dx.doi.org/10.1016/j.jnt.2003.06.007
 [H]
Christopher
Hooley, On Artin’s conjecture, J. Reine Angew. Math.
225 (1967), 209–220. MR 0207630
(34 #7445)
 [MAv]
Pieter
Moree, On the average number of elements in a finite field with
order or index in a prescribed residue class, Finite Fields Appl.
10 (2004), no. 3, 438–463. MR 2067608
(2005f:11219), http://dx.doi.org/10.1016/j.ffa.2003.10.001
 [MFi]
Pieter
Moree, Convoluted convolved Fibonacci numbers, J. Integer Seq.
7 (2004), no. 2, Article 04.2.2, 14 pp. (electronic).
MR
2084694 (2005i:11021)
 [MWi]
Pieter
Moree, The formal series Witt transform, Discrete Math.
295 (2005), no. 13, 143–160. MR 2143453
(2006b:05015), http://dx.doi.org/10.1016/j.disc.2005.03.004
 [M0]
P. Moree, On primes for which divides ord, Funct. Approx. Comment. Math. 33 (2005), 8595.
 [M1]
Pieter
Moree, On the distribution of the order and index of
𝑔\𝑝𝑚𝑜𝑑𝑝 over residue
classes. I, J. Number Theory 114 (2005), no. 2,
238–271. MR 2167970
(2006e:11152), http://dx.doi.org/10.1016/j.jnt.2004.09.004
 [M2]
Pieter
Moree, On the distribution of the order and index of
𝑔\𝑝𝑚𝑜𝑑𝑝 over residue
classes. II, J. Number Theory 117 (2006), no. 2,
330–354. MR 2213769
(2007b:11150), http://dx.doi.org/10.1016/j.jnt.2005.06.006
 [M3]
P. Moree, On the distribution of the order and index of over residue classes III, arXiv:math.NT/0405527, Journal of Number Theory, to appear.
 [P]
F.
Pappalardi, On Hooley’s theorem with weights, Rend. Sem.
Mat. Univ. Politec. Torino 53 (1995), no. 4,
375–388. Number theory, II (Rome, 1995). MR 1452393
(98c:11102)
 [W1]
K.
Wiertelak, On the density of some sets of primes. IV, Acta
Arith. 43 (1984), no. 2, 177–190. MR 736730
(86e:11081)
 [W2]
Kazimierz
Wiertelak, On the density of some sets of primes 𝑝, for
which
𝑛\𝑚𝑖𝑑{𝑜𝑟𝑑}_{𝑝}𝑎,
Funct. Approx. Comment. Math. 28 (2000), 237–241.
Dedicated to Włodzimierz Staś on the occasion of his 75th
birthday. MR
1824009 (2003a:11120)
 [Z]
D. Zagier, personal communication.
 [CM]
 K. Chinen and L. Murata, On a distribution property of the residual order of , I, II, J. Number Theory 105 (2004), 6081, 82100. MR 2032442 (2005c:11117); MR 2032443 (2005c:11118)
 [H]
 C. Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209220. MR 0207630 (34:7445)
 [MAv]
 P. Moree, On the average number of elements in a finite field with order or index in a prescribed residue class, Finite Fields Appl. 10 (2004), 438463. MR 2067608 (2005f:11219)
 [MFi]
 P. Moree, Convoluted convolved Fibonacci numbers, J. Integer Seq. 7 (2004), Article 04.2.2, 16 pp. (electronic). MR 2084694 (2005i:11021)
 [MWi]
 P. Moree, The formal series Witt transform, Discrete Math. 295 (2005), 143160. MR 2143453 (2006b:05015)
 [M0]
 P. Moree, On primes for which divides ord, Funct. Approx. Comment. Math. 33 (2005), 8595.
 [M1]
 P. Moree, On the distribution of the order and index of over residue classes I, J. Number Theory 114 (2005), 238271. MR 2167970 (2006e:11152)
 [M2]
 P. Moree, On the distribution of the order and index of over residue classes II, J. Number Theory 117 (2006), 330354. MR 2213769
 [M3]
 P. Moree, On the distribution of the order and index of over residue classes III, arXiv:math.NT/0405527, Journal of Number Theory, to appear.
 [P]
 F. Pappalardi, On Hooley's theorem with weights, Number theory, II (Rome, 1995), Rend. Sem. Mat. Univ. Politec. Torino 53 (1995), 375388. MR 1452393 (98c:11102)
 [W1]
 K. Wiertelak, On the density of some sets of primes, IV, Acta Arith. 43 (1984), 177190. MR 0736730 (86e:11081)
 [W2]
 K. Wiertelak, On the density of some sets of primes , for which , Funct. Approx. Comment. Math. 28 (2000), 237241. MR 1824009 (2003a:11120)
 [Z]
 D. Zagier, personal communication.
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Additional Information
Pieter Moree
Affiliation:
MaxPlanckInstitut für Mathematik, Vivatsgasse 7, D53111 Bonn, Germany
Email:
moree@mpimbonn.mpg.de
DOI:
http://dx.doi.org/10.1090/S1079676206001685
PII:
S 10796762(06)001685
Received by editor(s):
February 5, 2006
Published electronically:
August 18, 2006
Communicated by:
Brian Conrey
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
