Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions

James Haglund

Open Mathematics (2011)

  • Volume: 9, Issue: 2, page 302-318
  • ISSN: 2391-5455

Abstract

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Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡N(z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡N(z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.

How to cite

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James Haglund. "Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions." Open Mathematics 9.2 (2011): 302-318. <http://eudml.org/doc/269589>.

@article{JamesHaglund2011,
abstract = {Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡N(z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡N(z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.},
author = {James Haglund},
journal = {Open Mathematics},
keywords = {Riemann Hypothesis; Incomplete Gamma function; The Riemann Hypothesis; incomplete gamma function},
language = {eng},
number = {2},
pages = {302-318},
title = {Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions},
url = {http://eudml.org/doc/269589},
volume = {9},
year = {2011},
}

TY - JOUR
AU - James Haglund
TI - Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions
JO - Open Mathematics
PY - 2011
VL - 9
IS - 2
SP - 302
EP - 318
AB - Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡N(z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡N(z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.
LA - eng
KW - Riemann Hypothesis; Incomplete Gamma function; The Riemann Hypothesis; incomplete gamma function
UR - http://eudml.org/doc/269589
ER -

References

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  10. [10] Ki H., On the zeros of approximations of the Ramanujan ≡-function, Ramanujan J., 2008, 17(1), 123–143 http://dx.doi.org/10.1007/s11139-007-9046-4 Zbl1238.11080
  11. [11] Mahler K., Über die Nullstellen der unvollständigen Gammafunktionen, Rend. Circ. Mat. Palermo, 1930, 54, 1–31 http://dx.doi.org/10.1007/BF03021175 Zbl56.0310.01
  12. [12] Nielsen N., Die Gammafunktion, Chelsea Publishing Co., New York, 1965 
  13. [13] Pólya G., Bemerkung über die Integraldarstellung der Riemannschen ζ-Funktion, Acta Math., 1926, 48(3–4), 305–317 http://dx.doi.org/10.1007/BF02565336 
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