A Proof of the Riemann Hypothesis

Young Hwan Yun^1,

¹Zero Theoretical Physics Laboratory, Seoul, Republic of Korea

Cite this paper:
Young Hwan Yun. A Proof of the Riemann Hypothesis. American Journal of Applied Mathematics and Statistics. 2024; 12(4):86-92. doi: 10.12691/ajams-12-4-3

Abstract

This paper presents an intuitive method for proving the Riemann Hypothesis. It begins by deriving the relationship equation at the zeros of the Riemann zeta function from Riemann's functional equation. This equation follows the Schwarz reflection principle, indicating that the zeros of the zeta function are restricted to the line with a real part of 1/2 in the complex plane. Furthermore, using the Schwarz reflection principle, it concludes that zeros cannot exist outside the critical line. Therefore, the Riemann Hypothesis is true.

Keywords:
Zeta function trivial zeros critical line functional equation

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Figures

References:

[1]	Riemann, Bernhard. die Hypothesen, welche der Geometrie zu Grunde liegen.Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 13 (1861): 236-252.
[2]	Euler, Leonhard (1740). De Summis Serierum Reciprocarum. Commentarii academiae scientiarum Petropolitanae. Vol. 7, pp. 123-134.
[3]	Odlyzko, A. M. (1987), ‘On the distribution of spacings between zeros of the zeta function’, Mathematics of Computation 48(177), 273-308.
[4]	Czerwik, Stephan (2002). Functional Equations and Inequalities in Several Variables. P O Box 128, Farrer Road, Singapore 912805: World Scientific Publishing Co. p. 410. ISBN 981-02-4837-7.
[5]	A. (2003). The Riemann Zeta-Function: Theory and Applications. Dover Publications.
[6]	Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function. Oxford University Press.
[7]	Derbyshire, J. "The Prime Number Theorem." Ch. 3 Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics New York: Penguin, pp. 32-47, 2004.
[8]	https://brilliant.org/wiki/riemann-zeta-function.
[9]	Ahlfors, L. V. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill, 1979.
[10]	"Riemann-Schwarz principle", Encyclopedia of Mathematics, EMS Press, 2001 [1994].
[11]	E. C. Titchmarsh, "The Theory of the Riemann Zeta-Function", Oxford University Press; 2nd edition (February 5, 1987).
[12]	Eric Weisstein. Riemann Zeta Function Zeros. Retrieved 24 April 2021.
[13]	Bui, H. Q., & Keating, J. P. (2019). The first 10 billion zeros of the Riemann zeta function, and zeros computation at very large height. Research in the Mathematical Sciences, 6(1), 10.
[14]	Hardy, G. H.; Littlewood, J. E. (1921), The zeros of Riemann's zeta-function on the critical line, Math. Z., 10 (3-4): 283-317.
[15]	Alzer, H.Monotonicity Properties of the Riemann Zeta Function. Mediterr. J. Math. 9, 439–452 (2012).