Zeros of the dilogarithm
HTML articles powered by AMS MathViewer
- by Cormac O’Sullivan PDF
- Math. Comp. 85 (2016), 2967-2993 Request permission
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.References
- 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, DOI 10.1017/CBO9781107325937
- Miklós Bóna, Combinatorics of permutations, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2012. With a foreword by Richard Stanley. MR 2919720, DOI 10.1201/b12210
- 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, DOI 10.1090/S2330-1511-2014-00014-6
- B. Fornberg and K. S. Kölbig, Complex zeros of the Jonquière or polylogarithm function, Math. Comp. 29 (1975), 582–599. MR 369278, DOI 10.1090/S0025-5718-1975-0369278-0
- G. Frobenius, Über die Bernoulli’sehen Zahlen und die Eulerschen Polynome, Sitz. Berichte Preuss. Akad. Wiss. (1910), 808–847.
- W. Gawronski, On the zeros of power series with rational coefficients. III, Arch. Math. (Basel) 32 (1979), no. 4, 368–376. MR 545160, DOI 10.1007/BF01238513
- 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, DOI 10.1307/mmj/1029003073
- Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
- 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
- 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, DOI 10.5802/afst.173
- 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, DOI 10.1098/rspa.2003.1156
- C. O’Sullivan, Asymptotics for the partial fractions of the restricted partition generating function I. arXiv:1507.07975.
- C. O’Sullivan , Asymptotics for the partial fractions of the restricted partition generating function II. arXiv:1507.07977.
- Cormac O’Sullivan, On the partial fraction decomposition of the restricted partition generating function, Forum Math. 27 (2015), no. 2, 735–766. MR 3334080, DOI 10.1515/forum-2012-0073
- Alexander Peyerimhoff, On the zeros of power series, Michigan Math. J. 13 (1966), 193–214. MR 190304
- Hans Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman. MR 0364103, DOI 10.1007/978-3-642-80615-5
- 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, DOI 10.1080/10236198.2012.678837
- S. H. Siraždinov, The limit distribution of the roots of Euler polynomials, Dokl. Akad. Nauk SSSR 239 (1978), no. 1, 56–59 (Russian). MR 0476963
- S. L. Sobolev, Roots of Euler polynomials, Dokl. Akad. Nauk SSSR 235 (1977), no. 2, 277–280 (Russian). MR 0501764
- S. L. Sobolev, On extreme roots of Euler polynomials, Dokl. Akad. Nauk SSSR 242 (1978), no. 5, 1016–1019 (Russian). MR 510252
- S. L. Sobolev, More on roots of Euler polynomials, Dokl. Akad. Nauk SSSR 245 (1979), no. 4, 801–804 (Russian). MR 527714
- S. L. Sobolev, On the asymptotic behavior of Euler polynomials, Dokl. Akad. Nauk SSSR 245 (1979), no. 2, 304–308 (Russian). MR 526632
- 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, DOI 10.1007/s11075-007-9153-8
- Don Zagier, The dilogarithm function, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 3–65. MR 2290758, DOI 10.1007/978-3-540-30308-4_{1}
Additional Information
- Cormac O’Sullivan
- Affiliation: Department of Mathematics, The CUNY Graduate Center, New York, New York 10016-4309
- MR Author ID: 658848
- Email: cosullivan@gc.cuny.edu
- 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.
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 2967-2993
- MSC (2010): Primary 33B30, 30C15; Secondary 11P82
- DOI: https://doi.org/10.1090/mcom/3065
- MathSciNet review: 3522977