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Let $\phi(x)$ be the Euler function.

As presented in the Special values section there,

Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives $$\int_0^1 \phi(x) \, dx = \frac{8\sqrt{3/23}\pi\sinh(\sqrt{23}\pi/6)}{2\cosh(\sqrt{23}\pi/3)-1} \approx 1-0.631587464068566$$.

Can anybody assert if this result is actually correct, and provide a reference for it? As stated in the section,

This result needs concrete references, as I have been unable to verify it

projectilemotion
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    See https://math.stackexchange.com/questions/3019758/show-that-int-01-prod-n-geq-1-1-xn-dx-frac4-pi-sqrt3-sinh-fra?noredirect=1 or https://math.stackexchange.com/questions/1883140/an-integration-of-product-1-xn. The complex analytic method being referenced in the Wikipedia article is the well-known formula $$\sum_{k=-\infty}^{\infty}\left(-1\right)^{k}f\left(k\right)=-\sum\left{ \textrm{residues of }\pi\csc\left(\pi z\right)f(z)\textrm{ at }f\left(z\right)\textrm{'s poles}\right},$$ which follows from the residue theorem. – projectilemotion Jun 02 '21 at 14:01
  • The result given in both questions I linked can be shown to be equivalent to yours by some trigonometric identities. – projectilemotion Jun 02 '21 at 14:02
  • A correct/better version of this is (4) in Glasser, Some Integrals of the Dedekind $\eta$-Function, specialized to $y = 23/24$ plus a few identities. – Eric Towers Jun 02 '21 at 14:08
  • @projectilemotion That is a gorgeous formula! How have I not seen in before. – K.defaoite Jun 02 '21 at 15:08
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    @K.defaoite It's beautiful indeed! There are several assumptions that $f$ needs to satisfy which I forgot to mention above, see this article. There is a similar formula $$\sum_{k=-\infty}^{\infty} f\left(k\right)=-\sum\left{ \textrm{residues of }\pi\cot\left(\pi z\right)f(z)\textrm{ at }f\left(z\right)\textrm{'s poles}\right}.$$ From my experience, this formula is rarely taught in complex analysis courses. – projectilemotion Jun 02 '21 at 19:22

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