Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A NOTE ON THE ZEROS OF JENSEN POLYNOMIALS

  • Kim, Young-One (Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University) ;
  • Lee, Jungseob (Department of Mathematics Ajou University)
  • Received : 2021.10.08
  • Accepted : 2021.12.31
  • Published : 2022.07.01
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Abstract

Sufficient conditions for the Jensen polynomials of the derivatives of a real entire function to be hyperbolic are obtained. The conditions are given in terms of the growth rate and zero distribution of the function. As a consequence some recent results on Jensen polynomials, relevant to the Riemann hypothesis, are extended and improved.

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References

  1. M. Chasse, Laguerre multiplier sequences and sector properties of entire functions, Complex Var. Elliptic Equ. 58 (2013), no. 7, 875-885. https://doi.org/10.1080/17476933.2011.584250
  2. N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197-226. http://projecteuclid.org/euclid.dmj/1077476111 https://doi.org/10.1215/S0012-7094-50-01720-0
  3. W. Gontcharoff, Recherches sur les derivees successives des fonctions analytiques, Ann. Sci. Ecole Norm. Sup. (3) 47 (1930), 1-78. https://doi.org/10.24033/asens.798
  4. M. Griffin, K. Ono, L. Rolen, J. Thorner, Z. Tripp, and I. Wagner, Jensen polynomials for the Riemann xi function, preprint arXiv:1910.01227, (2020).
  5. M. Griffin, K. Ono, L. Rolen, and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA 116 (2019), no. 23, 11103-11110. https://doi.org/10.1073/pnas.1902572116
  6. H. Ki and Y.-O. Kim, De Bruijn's question on the zeros of Fourier transforms, J. Anal. Math. 91 (2003), 369-387. https://doi.org/10.1007/BF02788795
  7. Y.-O. Kim, Critical points of real entire functions and a conjecture of Polya, Proc. Amer. Math. Soc. 124 (1996), no. 3, 819-830. https://doi.org/10.1090/S0002-9939-96-03083-3
  8. Y.-O. Kim, Critical zeros and nonreal zeros of successive derivatives of real entire functions, J. Korean Math. Soc. 33 (1996), no. 3, 657-667.
  9. M.-H. Kim, Y.-O. Kim, and J. Lee, The convergence of a sequence of polynomials with restricted zeros, J. Math. Anal. Appl. 478 (2019), no. 2, 1121-1132. https://doi.org/10.1016/j.jmaa.2019.06.005
  10. N. Obrechkoff, Sur une generalisation du theoreme de Poulain et Hermite pour les zeros reels des polynomes reels, Acta Math. Acad. Sci. Hungar. 12 (1961), 175-184. https://doi.org/10.1007/BF02066679
  11. C. O'Sullivan, Zeros of Jensen polynomials and asymptotics for the Riemann xi function, Res. Math. Sci. 8 (2021), no. 3, Paper No. 46, 27 pp. https://doi.org/10.1007/s40687-020-00240-5
  12. D. Platt and T. Trudgian, The Riemann hypothesis is true up to 3 .1012, Bull. Lond. Math. Soc. 53 (2021), no. 3, 792-797. https://doi.org/10.1112/blms.12460
  13. G. Polya, Uber Annaherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279-295. https://doi.org/10.1007/BF03016033
  14. G. Polya, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178-191. https://doi.org/10.1090/S0002-9904-1943-07853-6
  15. J. Schur and G. Polya, Uber zwei Arten von Faktorenfolgen in der Theorie der alge-braischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89-113. https://doi.org/10.1515/crll.1914.144.8