Closed-form expression for $prod_{n=0}^{infty}frac{(4n+3)^{1/(4n+3)}}{(4n+5)^{1/(4n+5)}}$?

Question

I have recently come across this infinite product, and I was wondering what methods I could use to express the product in closed-form (if it is even possible):
$$prod_{n=0}^{infty}dfrac{(4n+3)^{1/(4n+3)}}{(4n+5)^{1/(4n+5)}}=dfrac{3^{1/3}}{5^{1/5}}cdot dfrac{7^{1/7}}{9^{1/9}}cdotdfrac{11^{1/11}}{13^{1/13}}cdotdfrac{15^{1/15}}{17^{1/17}}cdotdotsb$$
Thanks in advance!

Jack D'Aurizio · Answer

$$expsum_{ngeq 0}left(frac{log(4n+3)}{4n+3}-frac{log(4n+5)}{4n+5}right)=expleft[-sum_{ngeq 1}frac{chi_4(n)log(n)}{n}right]$$
equals
$$lim_{sto 1^+}expleft[frac{d}{ds}sum_{ngeq 1}frac{chi_4(n)}{n^s}right]=expbeta'(1)$$
with $beta$ being Dirichlet's beta function. According to equation (18) the closed form is
$$ exp,left[frac{pi}{4}left(gamma+logfrac{4pi^3}{Gammaleft(frac{1}{4}right)^4}right)right].$$
A derivation can be found here.

Closed-form expression for $prod_{n=0}^{infty}frac{(4n+3)^{1/(4n+3)}}{(4n+5)^{1/(4n+5)}}$?

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