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Joachim von zur Gathen; Arnold Knopfmacher; Florian Luca; Lutz G. Lucht; Igor E. Shparlinski
Average order in cyclic groups
Journal de théorie des nombres de Bordeaux, 16 no. 1 (2004), p. 107-123, doi: 10.5802/jtnb.436
Article PDF | Reviews MR 2145575 | Zbl 1079.11003 | 1 citation in Cedram
[1] T. M. Apostol (1976), Introduction to Analytic Number Theory. Springer-Verlag, New York. MR 434929 | Zbl 0335.10001
[2] P. T. Bateman (1972) . The distribution of values of the Euler function. Acta Arithmetica 21, 329–345.
Article | MR 302586 | Zbl 0217.31901
[3] C. K. Caldwell & Y. Gallot (2000), Some results for $n! \pm 1$ and $2 \cdot 3 \cdot 5 \cdots p \pm 1$. Preprint.
[4] J. R. Chen (1973), On the representation of a large even integer as a sum of a prime and a product of at most two primes. Scientia Sinica 16, 157–176. MR 434997 | Zbl 0319.10056
[5] P. D. T. A. Elliott (1985), Arithmetic functions and integer products, volume 272 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York. MR 766558 | Zbl 0559.10032
[6] K. Ford (1999), The number of solutions of $\phi (x)=m$. Annals of Mathematics 150, 1–29. MR 1715326 | Zbl 0978.11053
[7] H. Halberstam & H.E. Richert (1974), Sieve Methods. Academic Press. MR 424730 | Zbl 0298.10026
[8] G. H. Hardy & E. M. Wright (1962), An introduction to the theory of numbers. Clarendon Press, Oxford. 1st edition 1938. MR 67125 | Zbl 0020.29201
[9] K.-H. Indlekofer (1980), A mean-value theorem for multiplicative functions. Mathematische Zeitschrift 172, 255–271. MR 581443 | Zbl 0416.10035
[10] K.-H. Indlekofer (1981), Limiting distributions and mean-values of multiplicative arithmetical functions. Journal für die reine und angewandte Mathematik 328, 116–127. MR 636199 | Zbl 0455.10036
[11] W. Keller (2000). Private communication.
[12] D. G. Kendall & R. A. Rankin (1947), On the number of Abelian groups of a given order. Quarterly Journal of Mathematics 18, 197–208. MR 22569 | Zbl 0031.15303
[13] J. Knopfmacher (1972), Arithmetical properties of finite rings and algebras, and analytic number theory. II. Journal für die reine und angewandte Mathematik 254, 74–99. MR 364132 | Zbl 0246.10033
[14] J. Knopfmacher (1973), A prime divisor function. Proceedings of the American Mathematical Society 40, 373–377. MR 327694 | Zbl 0267.10059
[15] J. Knopfmacher & J. N. Ridley (1974), Prime-Independent Arithmetical Functions. Annali di Matematica 101(4), 153–169. MR 392872 | Zbl 0293.10026
[16] W. LeVeque (1977), Fundamentals of Number Theory. Addison-Wesley. MR 480290 | Zbl 0368.10001
[17] H. L. Montgomery (1970), Primes in arithmetic progressions. Michigan Mathematical Journal 17, 33–39.
Article | MR 257005 | Zbl 0209.34804
[18] H. L. Montgomery (1987), fluctuations in the mean of Euler’s phi function. Proceedings of the Indian Academy of Sciences (Mathematical Sciences) 97(1-3), 239–245. MR 983617 | Zbl 0656.10042
[19] A. G. Postnikov (1988), Introduction to analytic number theory. Volume 68 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI. MR 932727 | Zbl 0641.10001
[20] H. Riesel & R. C. Vaughan (1983), On sums of primes. Arkiv for Matematik 21(1), 46–74. MR 706639 | Zbl 0516.10044
[21] I. E. Shparlinski (1990), Some arithmetic properties of recurrence sequences. Matematicheskie Zametki 47(6), 124–131. (in Russian); English translation in Mathematical Notes 47, (1990), 612–617. MR 1074537 | Zbl 0714.11009
[22] P. J. Stephens (1969), An Average Result for Artin’s Conjecture. Mathematika 16(31), 178–188. MR 498449 | Zbl 0186.08402
[23] A. Walfisz (1963), Weylsche Exponentialsummen in der neueren Zahlentheorie. Number 15 in Mathematische Forschungsberichte. VEB Deutscher Verlag der Wissenschaften, Berlin. MR 220685 | Zbl 0146.06003
Joachim von zur Gathen; Arnold Knopfmacher; Florian Luca; Lutz G. Lucht; Igor E. Shparlinski
Average order in cyclic groups
Journal de théorie des nombres de Bordeaux, 16 no. 1 (2004), p. 107-123, doi: 10.5802/jtnb.436
Article PDF | Reviews MR 2145575 | Zbl 1079.11003 | 1 citation in Cedram
Résumé - Abstract
For each natural number $n$ we determine the average order $\alpha (n)$ of the elements in a cyclic group of order $n$. We show that more than half of the contribution to $\alpha (n)$ comes from the $\varphi (n)$ primitive elements of order $n$. It is therefore of interest to study also the function $\beta (n)=\alpha (n)/\varphi (n)$. We determine the mean behavior of $\alpha $, $\beta $, $1/\beta $, and also consider these functions in the multiplicative groups of finite fields.
Bibliography
[2] P. T. Bateman (1972) . The distribution of values of the Euler function. Acta Arithmetica 21, 329–345.
Article | MR 302586 | Zbl 0217.31901
[3] C. K. Caldwell & Y. Gallot (2000), Some results for $n! \pm 1$ and $2 \cdot 3 \cdot 5 \cdots p \pm 1$. Preprint.
[4] J. R. Chen (1973), On the representation of a large even integer as a sum of a prime and a product of at most two primes. Scientia Sinica 16, 157–176. MR 434997 | Zbl 0319.10056
[5] P. D. T. A. Elliott (1985), Arithmetic functions and integer products, volume 272 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York. MR 766558 | Zbl 0559.10032
[6] K. Ford (1999), The number of solutions of $\phi (x)=m$. Annals of Mathematics 150, 1–29. MR 1715326 | Zbl 0978.11053
[7] H. Halberstam & H.E. Richert (1974), Sieve Methods. Academic Press. MR 424730 | Zbl 0298.10026
[8] G. H. Hardy & E. M. Wright (1962), An introduction to the theory of numbers. Clarendon Press, Oxford. 1st edition 1938. MR 67125 | Zbl 0020.29201
[9] K.-H. Indlekofer (1980), A mean-value theorem for multiplicative functions. Mathematische Zeitschrift 172, 255–271. MR 581443 | Zbl 0416.10035
[10] K.-H. Indlekofer (1981), Limiting distributions and mean-values of multiplicative arithmetical functions. Journal für die reine und angewandte Mathematik 328, 116–127. MR 636199 | Zbl 0455.10036
[11] W. Keller (2000). Private communication.
[12] D. G. Kendall & R. A. Rankin (1947), On the number of Abelian groups of a given order. Quarterly Journal of Mathematics 18, 197–208. MR 22569 | Zbl 0031.15303
[13] J. Knopfmacher (1972), Arithmetical properties of finite rings and algebras, and analytic number theory. II. Journal für die reine und angewandte Mathematik 254, 74–99. MR 364132 | Zbl 0246.10033
[14] J. Knopfmacher (1973), A prime divisor function. Proceedings of the American Mathematical Society 40, 373–377. MR 327694 | Zbl 0267.10059
[15] J. Knopfmacher & J. N. Ridley (1974), Prime-Independent Arithmetical Functions. Annali di Matematica 101(4), 153–169. MR 392872 | Zbl 0293.10026
[16] W. LeVeque (1977), Fundamentals of Number Theory. Addison-Wesley. MR 480290 | Zbl 0368.10001
[17] H. L. Montgomery (1970), Primes in arithmetic progressions. Michigan Mathematical Journal 17, 33–39.
Article | MR 257005 | Zbl 0209.34804
[18] H. L. Montgomery (1987), fluctuations in the mean of Euler’s phi function. Proceedings of the Indian Academy of Sciences (Mathematical Sciences) 97(1-3), 239–245. MR 983617 | Zbl 0656.10042
[19] A. G. Postnikov (1988), Introduction to analytic number theory. Volume 68 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI. MR 932727 | Zbl 0641.10001
[20] H. Riesel & R. C. Vaughan (1983), On sums of primes. Arkiv for Matematik 21(1), 46–74. MR 706639 | Zbl 0516.10044
[21] I. E. Shparlinski (1990), Some arithmetic properties of recurrence sequences. Matematicheskie Zametki 47(6), 124–131. (in Russian); English translation in Mathematical Notes 47, (1990), 612–617. MR 1074537 | Zbl 0714.11009
[22] P. J. Stephens (1969), An Average Result for Artin’s Conjecture. Mathematika 16(31), 178–188. MR 498449 | Zbl 0186.08402
[23] A. Walfisz (1963), Weylsche Exponentialsummen in der neueren Zahlentheorie. Number 15 in Mathematische Forschungsberichte. VEB Deutscher Verlag der Wissenschaften, Berlin. MR 220685 | Zbl 0146.06003
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