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Zeros of Jensen polynomials and asymptotics for the Riemann xi function

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Abstract

The classical criterion of Jensen for the Riemann hypothesis is that all of the associated Jensen polynomials have only real zeros. We find a new version of this criterion, using linear combinations of Hermite polynomials, and show that this condition holds in many cases. Detailed asymptotic expansions are given for the required Taylor coefficients of the xi function at 1/2 as well as related quantities. These results build on those in the recent paper of Griffin, Ono, Rolen and Zagier.

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Notes

  1. This range was extended in [10, Cor. 1.3] and may be further extended to \(d \leqslant 9\times 10^{24}\) using [20].

References

  1. Craven, T., Csordas, G.: Jensen polynomials and the Turán and Laguerre inequalities. Pac. J. Math. 136(2), 241–260 (1989)

    Article  Google Scholar 

  2. Chasse, M.: Laguerre multiplier sequences and sector properties of entire functions. Complex Var. Elliptic Equ. 58(7), 875–885 (2013)

    Article  MathSciNet  Google Scholar 

  3. Csordas, G., Norfolk, T.S., Varga, R.S.: The Riemann hypothesis and the Turán inequalities. Trans. Am. Math. Soc. 296(2), 521–541 (1986)

    MATH  Google Scholar 

  4. Coffey, M.W.: Asymptotic estimation of \(\xi ^{(2n)}(1/2)\): on a conjecture of Farmer and Rhoades. Math. Comput. 78(266), 1147–1154 (2009)

    Article  MathSciNet  Google Scholar 

  5. Csordas, G., Varga, R.S.: Necessary and sufficient conditions and the Riemann hypothesis. Adv. Appl. Math. 11(3), 328–357 (1990)

    Article  MathSciNet  Google Scholar 

  6. Dimitrov, D.K., Cheikh, Y.B.: Laguerre polynomials as Jensen polynomials of Laguerre–Pólya entire functions. J. Comput. Appl. Math. 233(3), 703–707 (2009)

    Article  MathSciNet  Google Scholar 

  7. Dimitrov, D.K., Lucas, F.R.: Higher order Turán inequalities for the Riemann \(\xi \)-function. Proc. Am. Math. Soc. 139(3), 1013–1022 (2011)

    Article  Google Scholar 

  8. Farmer, David W.: Jensen polynomials are not a viable route to proving the Riemann hypothesis. arXiv:2008.07206

  9. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  10. Griffin, M., Ono, K., Rolen, L., Thorner, J., Tripp, Z. Wagner I.: Jensen polynomials for the Riemann xi function. arXiv:1910.01227

  11. Griffin, M., Ono, K., Rolen, L., Zagier, D.: Jensen polynomials for the Riemann zeta function and other sequences. Proc. Natl. Acad. Sci. USA 116(23), 11103–11110 (2019)

    Article  MathSciNet  Google Scholar 

  12. Hoorfar, A., Hassani, M.: Inequalities on the Lambert \(W\) function and hyperpower function. J. Inequal. Pure Appl. Math. 9(2), 5 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Jensen, J.L.W.V.: Recherches sur la théorie des équations. Acta Math. 36(1), 181–195 (1913)

    Article  MathSciNet  Google Scholar 

  14. Ki, H., Kim, Y.-O.: On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture. Duke Math. J. 104(1), 45–73 (2000)

    Article  MathSciNet  Google Scholar 

  15. O’Sullivan, C.: A generalization of the Riemann–Siegel formula. arXiv:1811.01130

  16. O’Sullivan, C.: Revisiting the saddle-point method of Perron. Pac. J. Math. 298(1), 157–199 (2019)

    Article  MathSciNet  Google Scholar 

  17. Pólya, G.: Algebraische Untersuchungen über ganze Funktionen vom Geschlechte Null und Eins. J. Reine Angew. Math. 145, 224–249 (1915)

    MathSciNet  MATH  Google Scholar 

  18. Pólya, G.: Über die algebraisch-funktionentheoretischen Untersuchungen von J.L.W.V. Jensen. Kgl. Danske Vid. Sel. Math.-Fys.Medd. 7, 3–33 (1927)

    Google Scholar 

  19. Pólya, G., Schur, J.: Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914)

    MathSciNet  MATH  Google Scholar 

  20. Platt, D., Trudgian, T. : The Riemann hypothesis is true up to \(3 \cdot 10^{12}\). arXiv:2004.09765

  21. Pustyl’nikov, L.D.: Asymptotics of the coefficients of the Taylor series of the \(\xi (s)\) function. Izv. Ross. Akad. Nauk Ser. Mat. 65(1), 93–106 (2001)

    Article  MathSciNet  Google Scholar 

  22. Romik, D. : Orthogonal polynomial expansions for the Riemann xi function. arXiv:1902.06330

  23. Szegő, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society, Colloquium Publications, Providence (1975)

    MATH  Google Scholar 

  24. Turán, P.: To the analytical theory of algebraic equations. Bulgar. Akad. Nauk. Otd. Mat. Fiz. Nauk. Izv. Mat. Inst., 3:123–137, 1959. Reprinted in the Collected Papers of Paul Turán, Ed. P. Erdős, Vol. 2, pp. 1080–1090. Akadémiai Kiadó (1990)

  25. Wagner, I.: The Jensen-Pólya program for various \(L\)-functions. Forum Math. 32(2), 525–539 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Thanks to Jacques Gélinas, Dan Romik, Tim Trudgian and both referees for their helpful comments.

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  1. Department of Mathematics, The CUNY Graduate Center, 365 Fifth Avenue, New York, NY, 10016-4309, USA

    Cormac O’Sullivan

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  1. Cormac O’Sullivan
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Correspondence to Cormac O’Sullivan.

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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

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O’Sullivan, C. Zeros of Jensen polynomials and asymptotics for the Riemann xi function. Res Math Sci 8, 46 (2021). https://doi.org/10.1007/s40687-020-00240-5

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