Irrationality of the Riemann-Zeta function at the positive integers

Yoochan Noh

doi:10.33043/8VNFBGQ6Fy

Authors

Yoochan Noh Korea International School

DOI:

https://doi.org/10.33043/8VNFBGQ6Fy

Abstract

The Riemann Zeta function, usually denoted by the Greek letter ζ , was defined in 1737 by a Swiss mathematician Leonhard Euler. This function is an infinite converging sum of powers of natural numbers, and it has explicit expressions in terms of π at positive even integers. In this paper we will discuss various irrationality proofs, focusing on irrationality of certain values of the Zeta function.

Author Biography

Yoochan Noh, Korea International School

Yoochan Noh is a high school student at Korea International School, with a serious focus on college-level mathematics subjects. After graduation, he plans to pursue degrees in Applied Mathematics and Computer Science, driven by his passion for problem-solving and the transformative potential of technology. Inspired by previous small individual research projects, Yoochan sought to undertake a more substantial and profound research endeavor in mathematics. This led him to explore the complex and deep theme of the Riemann-Zeta function. Yoochan’s aspiration extends beyond this research, as he hopes to engage in various types of research in the future, further expanding his understanding and contributing to the advancement of mathematical knowledge.

References

R. Apéry, Irrationalité de ζ 2 et ζ 3, Journées Arithmétiques de Luminy, Astérisque, no. 61 (1979), pp. 11-13.

T.M. Apostol, Introduction to analytic number theory, Springer Science & Business Media (1998).

F. Beukers, A note on the irrationality of ζ (2) and ζ (3), In: Pi: A Source Book. Springer, New York, NY (2004).

K. Conrad, Irrationality of π and e (2009), K.Conrad’s website: https://kconrad.math.uconn.edu/blurbs/analysis/irrational.pdf.

G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, Oxford university press (1979).

A.J. Hildebrand, Introduction to analytic number theory, Math 531 lecture notes (2005), University of Illinois website: https://faculty.math.illinois.edu/ hildebr/ant/.

S. Johannes, Infinitely many odd zeta values are irrational. By elementary means, arXiv preprint arXiv:1802.09410 (2018).

V.J. Kanovei, The correctness of Euler’s method for the factorization of the sine function into an infinite product, Russian Mathematical Surveys Volume 43 (1998).

D.H. Mayer, On a ζ function related to the continued fraction transformation, Bulletin de la Société Mathématique de France 104 (1976).

I. Niven, The transcendence of π, The American Mathematical Monthly 46.8 (1939): 469-471.

W. Rudin, Principles of mathematical analysis, McGraw-Hill New York (1976).

W.R. Sullivan, Numerous proofs of ζ (2) = (π^2)/6 (2013).

V.V. Zudilin, One of the eight numbers ζ (5), ζ (7),. . . , ζ (17), ζ (19) is irrational, Mathematical Notes 70.3 (2001): 426-431.

Irrationality of the Riemann-Zeta function at the positive integers

Authors

DOI:

Abstract

Author Biography

Yoochan Noh, Korea International School

References

Downloads

Published

How to Cite

Issue

Section

License