Algebra & Number Theory

On a problem of Arnold: The average multiplicative order of a given integer

Pär Kurlberg and Carl Pomerance

Full-text: Open access

Abstract

For coprime integers g and n, let g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of g(n) as nx 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 g(p) as px ranges over primes.

Article information

Source
Algebra Number Theory, Volume 7, Number 4 (2013), 981-999.

Dates
Received: 25 August 2011
Revised: 3 March 2012
Accepted: 24 May 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729988

Digital Object Identifier
doi:10.2140/ant.2013.7.981

Mathematical Reviews number (MathSciNet)
MR3095233

Zentralblatt MATH identifier
1282.11131

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions

Keywords
average multiplicative order

Citation

Kurlberg, Pär; Pomerance, Carl. On a problem of Arnold: The average multiplicative order of a given integer. Algebra Number Theory 7 (2013), no. 4, 981--999. doi:10.2140/ant.2013.7.981. https://projecteuclid.org/euclid.ant/1513729988


Export citation

References

  • V. Arnold, “Number-theoretical turbulence in Fermat–Euler arithmetics and large Young diagrams geometry statistics”, J. Mathematical Fluid Mech. 7:suppl. 1 (2005), S4–S50.
  • P. Erdős, C. Pomerance, and E. Schmutz, “Carmichael's lambda function”, Acta Arith. 58:4 (1991), 363–385.
  • C. Hooley, “On Artin's conjecture”, J. Reine Angew. Math. 225 (1967), 209–220.
  • P. Kurlberg, “On the order of unimodular matrices modulo integers”, Acta Arith. 110:2 (2003), 141–151.
  • P. Kurlberg and C. Pomerance, “On the periods of the linear congruential and power generators”, Acta Arith. 119:2 (2005), 149–169.
  • J. C. Lagarias and A. M. Odlyzko, “Effective versions of the Chebotarev density theorem”, pp. 409–464 in Algebraic number fields: $L$-functions and Galois properties (Durham, 1975), edited by A. Fröhlich, Academic Press, London, 1977.
  • S. Li and C. Pomerance, “On generalizing Artin's conjecture on primitive roots to composite moduli”, J. Reine Angew. Math. 556 (2003), 205–224.
  • F. Luca, “Some mean values related to average multiplicative orders of elements in finite fields”, Ramanujan J. 9:1-2 (2005), 33–44.
  • F. Luca and I. E. Shparlinski, “Average multiplicative orders of elements modulo $n$”, Acta Arith. 109:4 (2003), 387–411.
  • P. Moree and P. Stevenhagen, “A two-variable Artin conjecture”, J. Number Theory 85:2 (2000), 291–304.
  • F. Pappalardi, “On Hooley's theorem with weights”, Rend. Sem. Mat. Univ. Politec. Torino 53:4 (1995), 375–388.
  • I. E. Shparlinski, “On some dynamical systems in finite fields and residue rings”, Discrete Contin. Dyn. Syst. 17:4 (2007), 901–917.
  • P. J. Stephens, “Prime divisors of second-order linear recurrences, I”, J. Number Theory 8:3 (1976), 313–332.
  • S. S. Wagstaff, Jr., “Pseudoprimes and a generalization of Artin's conjecture”, Acta Arith. 41:2 (1982), 141–150.