Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2011 | 9 | 2 | 302-318
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
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.
  • [1] Apostol T.M., Modular Functions and Dirichlet Series in Number Theory, 2nd ed., Grad. Texts in Math., 41, Springer, New York, 1990
  • [2] Conrey J.B., More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. Reine Angew. Math., 1989, 399, 1–26
  • [3] Chudnovsky M., Seymour P., The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B, 2007, 97(3), 350–357
  • [4] Davenport H., Multiplicative Number Theory, 3rd ed., Grad. Texts in Math., 74, Springer, New York, 2000
  • [5] Edwards H.M., Riemann’s Zeta Function, reprint of the 1974 original, Dover Publications, Mineola, 2001
  • [6] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions. I, II, McGraw-Hill Book Company, New York-Toronto-London, 1953
  • [7] Gautschi W., The incomplete gamma functions since Tricomi, In: Tricomi’s Ideas and Contemporary Applied Mathematics, Rome/Turin, 1997, Atti Convegni Lincei, 147, Accad. Naz. Lincei, Rome, 1998, 203–237
  • [8] Gronwall T.H., Sur les zéros des fonctions P(z) et Q(z) associées à la fonction gamma, Ann. Sci. École Norm. Sup., 1916, 33, 381–393
  • [9] Hejhal D.A., On a result of G. Pólya concerning the Riemann ≡-function. J. Analyse Math., 1990, 55(1), 59–95
  • [10] Ki H., On the zeros of approximations of the Ramanujan ≡-function, Ramanujan J., 2008, 17(1), 123–143
  • [11] Mahler K., Über die Nullstellen der unvollständigen Gammafunktionen, Rend. Circ. Mat. Palermo, 1930, 54, 1–31
  • [12] Nielsen N., Die Gammafunktion, Chelsea Publishing Co., New York, 1965
  • [13] Pólya G., Bemerkung über die Integraldarstellung der Riemannschen ζ-Funktion, Acta Math., 1926, 48(3–4), 305–317
  • [14] Selberg A., On the zeros of Riemann’s zeta-function, Skr. Norske Vid. Akad. Oslo I., 1942, 10
  • [15] The Riemann Hypothesis, CMS Books Math., Springer, New York, 2008
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.