Riemann Hypothesis: Reflections on the Greatest Mystery of Mathematics
Abstract
The article presents a few aspects of the most important of the unsolved problems of mathematics, the Riemann hypothesis, in the context of the most recent results. The so-called Lehmer’s phenomenon discovered half a century ago and recently studied very intensively, as well as Newman’s hypothesis concerning the de Bruijn-Newman’s constant that is connected with it, seem to suggest that unexpectedly controversial elements sneak into mathematics, and they can be seen even on the level of the language used in it. This probably results from the fact that even the best tools that mathematicians at the moment have at their disposal and by means of which they try in vain to verify the Riemann hypothesis are – for fundamental reasons – inadequate.
References
Bell E.T.: Men of Mathematics, London: The Camelot Press Ltd. 1937.
Bombieri E.: I numeri portano a Dio, wywiad udzielony Giovanni Morandi dla „Il Giorno”, 11 marca 2003 r.
Csordas G., Odlyzko A.M., Smith W., Varga R.S.: A New Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant Λ, Electr. Trans. Num. Anal., 1 (1993), s. 104 111.
Drmota M.: Interview mit Andrew Odlyzko, „Internationale Mathematische Nachrichten”, Wien, August 1998, Nr. 178.
Euler L.: Variae observationes circa series infinitas. „Commentarii” 1737, 1744.
Lehmer D.H.: On the roots of the Riemann zeta function, „Acta Mathematica” 95 (1956), s. 291 298.
Maślanka K.: Pietro Mengoli i szeregi liczbowe. Prehistoria funkcji ζ Riemanna, „Kwartalnik Historii Nauki i Techniki”, 2004.
Maślanka K.: Riemann i jego funkcja ζ, „Zagadnienia Filozoficzne w Nauce” 30 (2002), s. 18-30.
Odlyzko A.M.: The 1020 th Zero of the Riemann Zeta Function and 70 Million of its Neighbors, AT&T Bell Laboratories preprint, 1997.
Riemann G.F.B.: Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte Königl. Preuss. Akad. Wiss. Berlin, 671, 1859.
Simmons G.: Calculus Gems, New York: McGraw Hill Inc. 1992.
Staruszkiewicz A.: Wstęp do Zapisków autobiograficznych A. Einsteina, Kraków: Wyd. „Znak” 1996.
Zagier D.: The first 50 million prime numbers, „Mathematical Intelligencer”, 0 (1977), s. 7-19.
Copyright (c) 2005 Roczniki Filozoficzne
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
