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Correction: An Application of Free Probability to Arithmetic Functions

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Abstract

In this paper, we study free probability on tensor product algebra \(\mathfrak {M} = M\,\otimes _{\mathbb {C}}\,{\mathcal {A}}\) of a \(W^{*}\)-algebra M and the algebra \({\mathcal {A}}\), consisting of all arithmetic functions equipped with the functional addition and the convolution. We study free-distributional data of certain elements of \(\mathfrak {M}\), and study freeness on \(\mathfrak {M}\), affected by fixed primes.

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References

  1. Bump, D.: Automorphic Forms and Representations, Cambridge Studies in Adv. Math., 55, Cambridge Univ. Press. ISBN: 0-521-65818-7 (1996)

  2. Cho, I.: Operators induced by prime numbers. Methods Appl. Math. 19(4), 313–340 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Cho, I.: On dynamical systems induced by \(p\)-Adic number fields. Opuscula Math. 35(4), 445–484 (2015)

    Article  MathSciNet  Google Scholar 

  4. Cho, I.: Classification on arithmetic functions and corresponding free-moment \(L\)-functions. Bull. Korean Math. Soc. 52(3), 717–734 (2015)

    Article  MathSciNet  Google Scholar 

  5. Cho, I.: Free distributional data of arithmetic functions and corresponding generating functions. Compl. Anal. Oper. Theo. (2013). doi:10.1007/s11785-013-0331-9

  6. Cho, I., Gillespie, T.: Arithmetic functions and corresponding free probability determined by primes (2013) (submitted to Rocky Mt. J. Math)

  7. Cho, I., Jorgensen, P.E.T.: Krein-space operators induced by dirichlet characters, special issues: Contemp. Math. Amer. Math. Soc.: Commutative and Noncommutative Harmonic Analysis and Applications, pp. 3–33 (2014)

  8. Cho, I., Jorgensen, P.E.T.: Krein-space representations of arithmetic functions determined by primes. Alg. Rep. Theo. 17(6), 1809–1841 (2014)

    Article  MathSciNet  Google Scholar 

  9. Gillespie, T.: Superposition of zeroes of automorphic \(L\)-functions and functoriality, Univ. of Iowa. PhD Thesis (2010)

  10. Gillespie, T.: Prime number theorems for rankin-Selberg \(L\)-functions over number fields. Sci. China Math. 54(1), 35–46 (2011)

    Article  MathSciNet  Google Scholar 

  11. Radulescu, F.: Random matrices, amalgamated free products and subfactors of the \(C^{*}\)-algebra of a free group of nonsingular index. Invent. Math. 115, 347–389 (1994)

    Article  MathSciNet  Google Scholar 

  12. Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Am. Math. Soc. Mem. 132(627) (1998)

    Article  MathSciNet  Google Scholar 

  13. Voiculescu, D., Dykemma, K., Nica, A.: Free random variables. CRM Monograph Series, vol. 1. ISBN:978-08-21869703, American Mathematical Society (1992)

  14. Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: \(p\)-Adic Analysis and Mathematical Physics. Ser. Soviet & East European Math., vol 1. ISBN:978-981-02-0880-6, World Scientific (1994)

  15. Bach, E., Shallit, J.: Algorithmic Number Theory (Vol I), MIT Press Ser. Foundation. Comput. ISBN:0-262-02405-5, Published by MIT Press (1996)

  16. Ford, K.: The number of solutions of \(\phi (x)=m\). Ann. Math. 150(1), 283–311 (1999)

    Article  MathSciNet  Google Scholar 

  17. Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers (5-th Ed.). ISBN:978-0-19-853171-5, Published by Oxford Univ. Press (1980)

  18. Pettofrezzo, A.J., Byrkit, D.R.: Elements of Number Theory, LCCN: 77-81766, Published by Prentice Hall (1970)

  19. Lagarias, J.C.: Euler Constant: Euler’s Work and Modern Development, Bull (New Series). Am. Math. Soc. 50(4), 527–628 (2013)

    Article  Google Scholar 

  20. Blackadar, B.: Operator Algebras: Theory of \(C^{*}\)-Algebras and von Neumann Algebras. ISBN:978-3-540-28517-5, Published by Springer-Verlag (1965)

  21. Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. (2) 67, 325–327 (1958)

    Article  MathSciNet  Google Scholar 

  22. Voiculescu, D.: On a trace formula of M. G. Kreuin, operators in indefinite metric spaces, scattering theory and other topics (Bucharest, 1985). Oper. Theo, Adv. Appl., 24, 329–332 (1987)

  23. Voiculescu, D.: Symmetries of some reduced free product \(C^{*}\)-Algebras, operator algebras and their connections with topology and ergodic theory (Buchsteni, 1983). Lect. Notes Math. 1132, 556–588 (1985)

  24. Alpay, D., Jorgensen, P.E.T., Salomon, G.: On Free Stochastic Processes and Their Derivatives. (2013) (submitted). arXiv: 1311.3239

  25. Popescu, I.: Local functional inequalities in one-dimensional free probability. J. Funct. Anal. 264(6), 1456–1479 (2013)

    Article  MathSciNet  Google Scholar 

  26. Valentin, I.: A note on amalgamated boolean, orthogonal and conditionally monotone or anti-monotone products of operator-valued \(C^{*}\)-algebraic probability spaces. Rev. Roumaine Math. Pures Appl. 57(4), 341–370 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Williams, J.D.: A Khintchine decomposition for free probability. Ann. Probab. 40(5), 2236–2263 (2012)

    Article  MathSciNet  Google Scholar 

  28. Blitvic, N.: The \((q, t)\)-Gaussian process. J. Funct. Anal. 263(10), 3270–3305 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors sincerely apologize to the readers, and they hope this corrected version will be a chance to help fixing their mistakes with apology. The authors found mistakes in Sects. 4 and 5 in the original published paper thanks to Prof. Brent Nelson, who indicated the mistakes which the corresponding author made. The authors specially thank Prof. Nelson for his kind indication and opinions. Also, the authors apologize to the editors of the journal, Compl. Anal. Oper. Theo.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, St. Ambrose University, 421 Ambrose Hall, 518 W. Locust St., Davenport, IA, 52803, USA

    Ilwoo Cho & Palle E. T. Jorgensen

  2. Department of Mathematics and Statistics, University of Iowa, 14 McLean Hall, Iowa City, IA, 52242, USA

    Ilwoo Cho & Palle E. T. Jorgensen

Authors
  1. Ilwoo Cho
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  2. Palle E. T. Jorgensen
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Correspondence to Ilwoo Cho.

Additional information

Communicated by Daniel Aron Alpay.

Communicated with Prof. Brent Nelson.

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Cho, I., Jorgensen, P.E.T. Correction: An Application of Free Probability to Arithmetic Functions. Complex Anal. Oper. Theory 12, 1567–1608 (2018). https://doi.org/10.1007/s11785-016-0604-x

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