Asymptotic formula for $\prod_{k=1}^{\infty}\zeta (2kn)$

Question

Suppose $n\geq 1$ is a positive integer. Can we find an asymptotic formula for this product below.

$$\prod_{k=1}^{\infty}\zeta (2kn)=\zeta (2n)\zeta (4n)\zeta (6n) \cdots$$

I tried to use $\zeta (2n)=\dfrac{(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}$ but couldn't get anywhere.

Answer 1

This is not an answer since just based on numerical simulation.

Being stuck with any formal approach I tried, I just used numerical simulation and what I observed is that, if $$P_n=\prod_{k=1}^{\infty}\zeta (2kn)$$ then $\log(P_n-1)$ varies as a linear function of $n$. A basic linear regression $(1\leq n\leq 50)$ gave $$\log(P_n-1)=0.135417-1.390256 n$$ both parameters being statistically significant.

Edit

Inspired by Igor Rivin's and Raymond Manzoni's answers.

Limiting the $\zeta$ function to the first term $(\zeta(s)=1+\frac 1 {2^s})$, it seems that, for large $n$, we can approximate $$P_n\approx 1+\frac 1 {4^n}$$ since $$\log(P_n)=\sum_{k=1}^\infty \log(\zeta(2kn))\approx\sum_{k=1}^\infty \log(1+\frac 1{2^{2kn}})\approx\sum_{k=1}^\infty \frac 1{2^{2kn}}=\frac{1}{4^n-1}$$ $$P_n\approx \exp(\frac{1}{4^n-1})\approx 1+\frac{1}{4^n}$$ $$\log(P_n-1)\approx -n \log(4)$$ ($\log(4)\approx 1.38629$ being quite close to the $1.390256$ from the empirical calculation).

Answer 2

For a reasonably large $n$ (that is, bigger than about $3$), $\zeta(n) \approx 1+ \frac{1}{2^n},$ which will give you the asymptotics you crave.

Answer 3

Let's examine the most significant terms of the product for $n\gg 1$ : \begin{align} \tag{1}P_n&:=\prod_{k=1}^{\infty}\zeta (2kn)\\ &=\zeta (2n)\;\zeta (4n)\cdots\\ \tag{*}&=(1+2^{-2n}+3^{-2n}+4^{-2n}+o(4^{-2n}))\;(1+2^{-4n}+o(4^{-2n}))\;(1+o(4^{-2n}))\\ \tag{2}P_n&=1+2^{-2n}+3^{-2n}+2\cdot 4^{-2n}+o(4^{-2n})\\ \end{align}

(the product of the remaining terms $\,\zeta(6n)\,\zeta(8n)\cdots$ in $(*)$ is rewritten $(1+o(4^{-2n}))$ since the most significant term (except $1$) is $\;2^{-6n}+2^{-8n}+\cdots=\dfrac {4^{-2n}}{4^n-1}$)

From $(2)$ we deduce the simple : $\boxed{\displaystyle P_n-1\sim 4^{-n}}\;$ which could be obtained with Igor Rivin's hint
(the coefficient $-1.390256$ in Claude's approximation is near $-\log(4)$, the constant term should disappear for large $n$...)

Concerning $P_n$ for small values of $n$ : $P_1=C_2$ was considered in :

• Bernd Kellner : "On asymptotic constants related to products of Bernoulli numbers and factorials" and at bernoulli.org
  ($C_2$ is written as an infinite product of Dedekind eta function there) ; concerning $C_1$ we have $\;C_1=\lim_{n\overset{>}{\to}\frac 12} (2n-1)P_n\;$ with $P_n$ becoming infinite as $n{\to}\frac 12$)
• Steven Finch : "Minkowski-Siegel Mass Constants" may also be of interest