This note proves that the first odd zeta value does not have a closed form\nformula $\\zeta(3)e r \\pi^3$ for any rational number $r \\in \\mathbb{Q}$.\nFurthermore, assuming the irrationality of the second odd zeta value\n$\\zeta(5)$, it is shown that $\\zeta(5)e r\...
The Zeta Quotient $\\zeta(3)/ \\pi^3$ is Irrational 来自 arXiv.org 喜欢 0 阅读量: 53 作者: NA Carella 摘要: This note proves that the first odd zeta value does not have a closed form formula $\\\zeta(3)e r \\\pi^3$ for any rational number $r \\\in \\\mathbb{Q}$...
Numerical evidence suggests that all values of corresponding to nontrivial zeros are irrational (e.g., Havil 2003, p. 195; Derbyshire 2004, p. 384). No known zeros with order greater than one are known. While the existence of such zeros would not disprove the Riemann hypothesis, it would...
It is named after Apéry, who proved in 1979 that ζ(3) is irrational (see [2]). For Re(z)>1 and q≠0,−1,−2,…, in 1882, Hurwitz [18] defined the partial zeta functionζ(z,q)=∑n=0∞1(n+q)z which generalized (1.1). As (1.1), this function can also be ...
Zudilin, W. "One of the Numbers , , , Is Irrational." Uspekhi Mat. Nauk 56, 149-150, 2001.Zvengrowski, P. and Saidak, F. "On the Modulus of the Riemann Zeta Function in the Critical Strip." Math. Slovaca 53, 145-172, 2003....
Our initial motivation to studyis a significant connection between the values ofat the natural numbers and the probabilitythatkrandom elements generate the group ring⟦⟧ofGoveras a⟦⟧-module. In this setting, we have the following theorem (see Sect.3). ...
so there is nothing more about their zeros. For rational values of a other than 1/2 or 1 or for transcendental values of a, it is known that ζ(z, a) has an infinity of zeros with Re z > 1. For irrational algebraic values of a it is not known whether there are any zeros with...
The thesis also looks briefly at the history of the subject of the irrationality of naturally occurring numbers in mathematics. The crux of all the proofs is the same and this is examined in detail.;The first proof that zeta (3) is irrational is taken from Van der Poorten's informal ...
Computer Science - Logic in Computer ScienceF.3This paper presents a complete formal verification of a proof that theevaluation of the Riemann zeta function at 3 is irrational, using the Coq proofassistant. This result was first presented by Ap\\'ery in 1978, and the proof wehave formalized...
In particular, at least one of ζ 2 (7) , ζ 2 (9) , ζ 2 (1 1) , ζ 2 (1 3) is irrational. Our approach is inspired by the recent work of Sprang. We construct explicit rational functions. The Volkenborn integrals of these rational functions' (higher-order) derivatives produce...