# A new proof that $\zeta(2n+1)$ for integer $n>1$ is irrational?

I recently came across this paper on ArXiv by N. A. Carella (an author with over 60 publications on the aforementioned website), published on 4$^{\rm th}$ June 2018, containing the following Theorem:

Theorem 1. For each fixed odd integer $s = 2k + 1 ≥ 3$, the zeta constant $ζ(s)$ is an irrational number.

Obviously purported "proofs" of the irrationality of the Riemann Zeta function at positive odd integers can be found in abundance, however given the reputation of the author perhaps this deserves a better look. On the bottom of page 4, he outlines his proof:

A different technique using two independent infinite sequences of rational approximations of the two constants $ζK(s)$, and $1/L(s, χ)$, which are linearly independent over the rational numbers, will be used to construct an infinite sequence of rational approximations for the zeta constant $ζ(2n + 1), n ≥ 1$. The properties of these sequences, such as sufficiently fast rates of convergence, are then used to derive the irrationality of any zeta constant $ζ(2n + 1), n ≥ 1$.

This seems like a plausible "angle of attack" since $\zeta(3)$ was also proved to be irrational by constructing a sequence of rational approximations that approximate the constant "too well" for it to be rational. Has this paper been reviewed somewhere or does someone (more educated than me) care to comment on the validity of this "proof"?

• IMO: If you look at the revision history, you should wait a few months and look for a newer version. – gammatester Jun 21 '18 at 14:34
• This is in the GM section of the maths arxiv. – Lord Shark the Unknown Jun 21 '18 at 14:56
• I am rather skeptical about the validity of such proof. A truncated $L$-function is not enough to prove the irrationality of $\zeta(3)$, either, Apery had to use an accelerated version, namely $\frac{5}{2}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\binom{2n}{n}}$. Carella's proof of the irrationality of $\pi e$ looks very fishy, too. – Jack D'Aurizio Jun 21 '18 at 17:05