An arithmetic function arising from Carmichael’s conjecture
Florian Luca[1]; Paul Pollack[2]
- [1] Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
- [2] University of Illinois at Urbana-Champaign Department of Mathematics Urbana, Illinois 61801, USA
Journal de Théorie des Nombres de Bordeaux (2011)
- Volume: 23, Issue: 3, page 697-714
- ISSN: 1246-7405
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top- R. C. Baker and G. Harman, Shifted primes without large prime factors. Acta Arith. 83 (1998), 331–361. Zbl0994.11033MR1610553
- W. D. Banks, J. B. Friedlander, F. Luca, F. Pappalardi, and I. E. Shparlinski, Coincidences in the values of the Euler and Carmichael functions. Acta Arith. 122 (2006), 207–234. Zbl1195.11128MR2239915
- R. D. Carmichael, On Euler’s -function. Bull. Amer. Math. Soc. 13 (1907), 241–243. Zbl38.0236.01MR1558451
- —, Note on Euler’s -function. Bull. Amer. Math. Soc. 28 (1922), 109–110. Zbl48.0158.04MR1560520
- N. G. de Bruijn, Asymptotic methods in analysis. Third ed., Dover Publications Inc., New York, 1981. Zbl0556.41021MR671583
- R. E. Dressler, A density which counts multiplicity. Pacific J. Math. 34 (1970), 371–378. Zbl0181.05302MR271057
- P. Erdős, On the normal number of prime factors of and some related problems concerning Euler’s -function. Quart. J. Math. 6 (1935), 205–213. Zbl0012.14905
- —, Some remarks on Euler’s -function and some related problems. Bull. Amer. Math. Soc. 51 (1945), 540–544. Zbl0061.08005MR12634
- P. Erdős, A. Granville, C. Pomerance, and C. Spiro, On the normal behavior of the iterates of some arithmetic functions. Analytic number theory (Allerton Park, IL, 1989), Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 165–204. Zbl0721.11034MR1084181
- P. Erdős and J.-L. Nicolas, Sur la fonction: nombre de facteurs premiers de . Enseign. Math. (2) 27 (1981), 3–27. Zbl0466.10037MR630957
- P. Erdős and C. Pomerance, On the normal number of prime factors of . Rocky Mountain J. Math. 15 (1985), 343–352. Zbl0617.10037MR823246
- K. Ford, The distribution of totients. Ramanujan J. 2 (1998), 67–151. Zbl0914.11053MR1642874
- —, The number of solutions of . Ann. of Math. (2) 150 (1999), 283–311. MR1715326
- A. Granville, Smooth numbers: computational number theory and beyond. Algorithmic number theory: lattices, number fields, curves and cryptography, Math. Sci. Res. Inst. Publ., vol. 44, Cambridge Univ. Press, Cambridge, 2008, pp. 267–323. Zbl1230.11157MR2467549
- G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number . Quart. J. Math. 58 (1917), 76–92. Zbl46.0262.03
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Fifth ed., Oxford University Press, New York, 1979. Zbl1159.11001MR568909
- F. Luca and C. Pomerance, On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions and . Colloq. Math. 92 (2002), 111–130. Zbl1027.11007MR1899242
- —, Irreducible radical extensions and Euler-function chains. Combinatorial number theory, de Gruyter, Berlin, 2007, pp. 351–361. MR2337059
- P. Pollack, On the greatest common divisor of a number and its sum of divisors. Michigan Math. J. 60 (2011), 199–214. Zbl1286.11152MR2785871
- C. Pomerance, Popular values of Euler’s function. Mathematika 27 (1980), 84–89. Zbl0437.10001MR581999
- —, On the distribution of pseudoprimes. Math. Comp. 37 (1981), 587–593. Zbl0511.10002MR628717
- —, Two methods in elementary analytic number theory. Number theory and applications (Banff, AB, 1988), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 135–161. MR1123073
- G. Robin, Estimation de la fonction de Tchebychef sur le -ième nombre premier et grandes valeurs de la fonction nombre de diviseurs premiers de . Acta Arith. 42 (1983), 367–389. Zbl0475.10034MR736719