Proof of a closed-form of ∏ n = 1 ∞ 1 e ( 1 + 1 3 n ) 3 n + 1 / 2

Question

Proof of a closed-form of       ∏          n      =      1              ∞            1    e              (      1      +              1                  3          n                    )              3      n      +      1              /            2

zalitiaf · Accepted Answer

The   log of your product is  =      ∑          n      ≥      1        −  1  +  (  3  n  +  1  )  log  ⁡  (  3  n  +  1  )  −  3  n  log  ⁡  3  n  −      1    2    log  ⁡  (  3  n  +  1  )  −      1    2    log  ⁡  (  3  n  )Up to a non-complicated constant coming from the regularization terms this is a mix of   ζ  (  −  1  )  ,      ζ    ′    (  −  1  )  ,  ζ  (  0  )  ,      ζ    ′    (  0  )  ,  log  ⁡  Γ  (  1      /    3  )  ,      L    ′    (  −  1  ,      χ    3    )  χ is a an odd Dirichlet character so   L  (  −  1  ,      χ    3    )  =  0 and the functional equation relates       L    ′    (  −  1  ,      χ    3    ) with   L  (  2  ,      χ    3    ), it doesn&#039;t have a closed-form because       χ    3   is odd, which is where your       ψ    1    (  1      /    3  ) term comes from

Proof of a closed-form of <munderover> ∏ <mrow class="MJX-TeXAtom-ORD">

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