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More Zeros of the Derivatives of the Riemann Zeta Function on the Left Half Plane

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Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 64))

Abstract

We present the discovery of many previously unknown zeros of the derivatives, ζ (k)(σ + it), of the Riemann zeta function for k ≤ 28 with \(-10 <\sigma < \frac{1} {2}\) and − 10 < t < 10. Each zero found was simple and our computations show an interesting behavior of the zeros of ζ (k), namely they seem to lie on curves which are extensions of certain chains of zeros of ζ (k) that were observed on the right half plane.

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References

  1. Apostol, T.M.: Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44(169), 223–232 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Binder, T., Pauli, S., Saidak, F.: New zero free regions for the derivatives of the Riemann Zeta function, Rocky Mountain Journal of Mathematics (2011)

    Google Scholar 

  3. Cohen, H., Villegas, F.R., Zagier, D.: Convergence acceleration of alternating series. Exp. Math. 9, 3–12 (2000)

    Article  MATH  Google Scholar 

  4. Edwards, H.M.: Riemann’s Zeta function. In: Pure and Applied Mathematics, vol. 58. Academic, New York (1974)

    Google Scholar 

  5. Janjic, M.: On non-central stirling numbers of the first kind. http://adsabs.harvard.edu/abs/2009arXiv0901.2655J (2009)

  6. Joansson, F., et al.: Mpmath: a Python library for arbitrary-precision floating-point arithmetic. http://code.google.com/p/mpmath/ (2010)

  7. Jungman, G., Gough, B., et al.: GSL - GNU Scientific Library. http://www.gnu.org/software/gsl/ (2011)

  8. Levinson, N., Montgomery, H.L.: Zeros of the derivatives of the Riemann Zeta function. Acta Math. 133, 49–65 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Skorokhodov, S.L.: Padé approximants and numerical analysis of the Riemann Zeta function. Zh. Vychisl. Mat. Mat. Fiz. 43(9), 1330–1352 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Speiser, A.: Geometrisches zur Riemannschen Zetafunktion. Math. Ann. 110, 514–521 (1934)

    Article  MathSciNet  Google Scholar 

  11. Spira, R.: Zero-free region for ζ (k)(s). J. Lond. Math. Soc. 40, 677–682 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Spira, R.: Another zero-free region for ζ (k)(s). Proc. Am. Math. Soc. 26(2), 246–247 (1970)

    MathSciNet  MATH  Google Scholar 

  13. Stein, W., et al.: Sage, Open-source Mathematics Software. http://www.sagemath.org (2012)

  14. Verma, D.P., Kaur, A.: Zero-free regions of derivatives of Riemann Zeta function. Proc. Indian Acad. Sci. Math. Sci. 91(3), 217–221 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yıldırım, C.Y.: Zeros of ζ ′ ′(s) and ζ ′ ′ ′(s) in σ < 1∕2. Turk. J. Math. 24(1), 89–108 (2000)

    Google Scholar 

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Correspondence to Ricky Farr .

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Farr, R., Pauli, S. (2013). More Zeros of the Derivatives of the Riemann Zeta Function on the Left Half Plane. In: Rychtář, J., Gupta, S., Shivaji, R., Chhetri, M. (eds) Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference. Springer Proceedings in Mathematics & Statistics, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9332-7_10

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