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Zeros of the dilogarithm


Author: Cormac O’Sullivan
Journal: Math. Comp. 85 (2016), 2967-2993
MSC (2010): Primary 33B30, 30C15; Secondary 11P82
DOI: https://doi.org/10.1090/mcom/3065
Published electronically: March 2, 2016
MathSciNet review: 3522977
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Abstract: We show that the dilogarithm has at most one zero on each branch, that each zero is close to a root of unity, and that they may be found to any precision with Newton's method. This work is motivated by applications to the asymptotics of coefficients in partial fraction decompositions considered by Rademacher. We also survey what is known about zeros of polylogarithms in general.


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  • [AAR99] George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • [Bón12] Miklós Bóna, Combinatorics of permutations, with a foreword by Richard Stanley, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2012. MR 2919720
  • [DG14] Michael Drmota and Stefan Gerhold, Disproof of a conjecture by Rademacher on partial fractions, Proc. Amer. Math. Soc. Ser. B 1 (2014), 121-134. MR 3280294, https://doi.org/10.1090/S2330-1511-2014-00014-6
  • [FK75] B. Fornberg and K. S. Kölbig, Complex zeros of the Jonquière or polylogarithm function, Math. Comp. 29 (1975), 582-599. MR 0369278 (51 #5513)
  • [Fro10] G. Frobenius, Über die Bernoulli'sehen Zahlen und die Eulerschen Polynome, Sitz. Berichte Preuss. Akad. Wiss. (1910), 808-847.
  • [Gaw79] W. Gawronski, On the zeros of power series with rational coefficients. III, Arch. Math. (Basel) 32 (1979), no. 4, 368-376. MR 545160 (81b:30011), https://doi.org/10.1007/BF01238513
  • [GS84] Wolfgang Gawronski and Ulrich Stadtmüller, On the zeros of Jonquière's function with a large complex parameter, Michigan Math. J. 31 (1984), no. 3, 275-293. MR 767608 (86a:30009), https://doi.org/10.1307/mmj/1029003073
  • [GKP94] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics: A foundation for computer science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. MR 1397498 (97d:68003)
  • [Jon89] A. Jonquière, Note sur la série $ \sum _{n=1}^{\infty } \frac {x^n}{n^s}$, Bull. Soc. Math. France 17 (1889), 142-152 (French). MR 1504064
  • [LR00] Édouard Le Roy, Sur les séries divergentes et les fonctions définies par un développement de Taylor, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 2 (1900), no. 3, 317-384 (French). MR 1508224
  • [Max03] Leonard C. Maximon, The dilogarithm function for complex argument, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2039, 2807-2819. MR 2015991 (2004h:33010), https://doi.org/10.1098/rspa.2003.1156
  • [O'Sa] C. O'Sullivan, Asymptotics for the partial fractions of the restricted partition generating function I. arXiv:1507.07975.
  • [O'Sb] C. O'Sullivan , Asymptotics for the partial fractions of the restricted partition generating function II. arXiv:1507.07977.
  • [O'S15] Cormac O'Sullivan, On the partial fraction decomposition of the restricted partition generating function, Forum Math. 27 (2015), no. 2, 735-766. MR 3334080, https://doi.org/10.1515/forum-2012-0073
  • [Pey66] Alexander Peyerimhoff, On the zeros of power series, Michigan Math. J. 13 (1966), 193-214. MR 0190304 (32 #7717)
  • [Rad73] Hans Rademacher, Topics in analytic number theory, edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. MR 0364103 (51 #358)
  • [SZ13] Andrew V. Sills and Doron Zeilberger, Rademacher's infinite partial fraction conjecture is (almost certainly) false, J. Difference Equ. Appl. 19 (2013), no. 4, 680-689. MR 3040823, https://doi.org/10.1080/10236198.2012.678837
  • [Sir78] S. H. Siraždinov, The limit distribution of the roots of Euler polynomials, Dokl. Akad. Nauk SSSR 239 (1978), no. 1, 56-59. MR 0476963 (57 #16508)
  • [Sob77] S. L. Sobolev, Roots of Euler polynomials, Dokl. Akad. Nauk SSSR 235 (1977), no. 2, 277-280. MR 0501764 (58 #19040)
  • [Sob78] S. L. Sobolev, On extreme roots of Euler polynomials, Dokl. Akad. Nauk SSSR 242 (1978), no. 5, 1016-1019. MR 510252 (82e:41049)
  • [Sob79a] S. L. Sobolev, More on roots of Euler polynomials, Dokl. Akad. Nauk SSSR 245 (1979), no. 4, 801-804. MR 527714 (80f:65052)
  • [Sob79b] S. L. Sobolev, On the asymptotic behavior of Euler polynomials, Dokl. Akad. Nauk SSSR 245 (1979), no. 2, 304-308. MR 526632 (81a:33022)
  • [Vep08] Linas Vepštas, An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions, Numer. Algorithms 47 (2008), no. 3, 211-252. MR 2385736 (2009e:33073), https://doi.org/10.1007/s11075-007-9153-8
  • [Zag07] Don Zagier, The dilogarithm function, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 3-65. MR 2290758 (2008h:33005), https://doi.org/10.1007/978-3-540-30308-4_1

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Additional Information

Cormac O’Sullivan
Affiliation: Department of Mathematics, The CUNY Graduate Center, New York, New York 10016-4309
Email: cosullivan@gc.cuny.edu

DOI: https://doi.org/10.1090/mcom/3065
Keywords: Dilogarithm zeros, Newton's method, polylogarithms
Received by editor(s): February 19, 2015
Published electronically: March 2, 2016
Additional Notes: Support for this project was provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.
Article copyright: © Copyright 2016 American Mathematical Society

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