DOI QR코드

DOI QR Code

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

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.

Keywords

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