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I have been working on a problem in number theory that I have reduced to the problem of showing that the two-term linear combination $$ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \text{Li}_2\left(2+\sqrt{3}\right)$$ of dilogarithmic values admits the following closed form: \begin{align*} & \frac{1}{96} \Bigg(29 \pi ^2-48 i \pi \cosh ^{-1}(7)+6 \Bigg(-4 \log ^2(2)+2 \log \left(2-\sqrt{3}\right) \log \left(10+4 \sqrt{6}\right) + \\ & \cos ^{-1}(2)^2-8 \log \left(\sqrt{2+\sqrt{3}}-1\right) \cosh ^{-1}(2)\Bigg)\Bigg). \end{align*} To clarify, using a basic linear transformation for the dilogarithm function, we can express the value $\text{Li}_2\left(2+\sqrt{3}\right)$, which does not correspond to a convergent series according to the usual definition for $\text{Li}_{2}$, in the following manner: $$ \text{Li}_2\left(\frac{1}{2+\sqrt{3}}\right) = -\text{Li}_2\left(2+\sqrt{3}\right)-\frac{\pi ^2}{6}-\frac{1}{2} \left(\log \left(2+\sqrt{3}\right)+i \pi \right)^2. $$ How can we prove the conjectured evaluation for $ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \text{Li}_2\left(2+\sqrt{3}\right)$ shown above? Intuitively, it seems that this should be manageable, but I feel stuck in terms of this problem. Below, I have outlined a few different approaches toward answering the question under consideration.

I have experimented with the use of Landen's identity and related identities concerning the Rogers $L$-function in order to rewrite the two-term linear combination under consideration, but this has not led me to any promising results. It might be worthwhile to consider using Legendre's chi-function or the inverse tangent integral, especially since this latter function is known to satisfy closed-form identities involving $2 + \sqrt{3}$ as an argument, as in the following: $$ 3 \text{Ti}_{2}\left( 2 + \sqrt{3} \right) = 2 \text{Ti}_{2}(1) + \frac{5}{4} \pi \log(2 + \sqrt{3}). $$ Perhaps these kinds of identities for $\text{Ti}_{2}\left( 2 + \sqrt{3} \right)$ could be manipulated so as to prove the desired evaluation for the two-term dilogarithm combination under consideration. What do you think?

John M. Campbell
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    Your conjecture is true, it's a bit more elegant to write in the following form $$\text{Li}_2(\frac{\sqrt{3}+2}{4})-8 \text{Li}_2\left(2-\sqrt{3}\right) = -\frac{\pi ^2}{4}-2 \log ^2(2)+\frac{5}{2} \log ^2\left(\sqrt{3}+2\right)-2 \log \left(\sqrt{3}+2\right) \log (2)$$ my approach is not straightforward. From which context of number theory did you encounter this problem? – pisco Apr 24 '22 at 16:51
  • @pisco Please show me your approach toward proving my conjecture. I would love to see your proof. As it turns out, I have shown how an open problem due to Zhi-Wei Sun is actually reducible to the problem of evaluating the two-term dilogarithm combination referred to above. This problem involves the sequence of alternating odd harmonic numbers. It would be great to collaborate with you on this. – John M. Campbell Apr 24 '22 at 16:57
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    I think I know which conjecture of Zhi-Wei Sun you're talking about. I solved many (not all) of those conjectures recently (not yet published), using theory of colored multiple zeta values (CMZV). Your question is actually closely related to CMZV: both sides of claimed equality is CMZV of level 12 and weight 2, any $\mathbb{Q}$-relations between them are completely known. This is my approach mentioned above. I think I also saw you before on ResearchGate? – pisco Apr 24 '22 at 17:01
  • @pisco Thanks for your response! I have been studying a number of papers on ResearchGate concerning CMZVs, but I do not have much experience concerning mathematical objects of this form. I have shown in an accessible way how Sun's conjecture reduces to the two-term dilogarithm combination I've been referring to, and if you would be willing to post your proof for this two-term $\text{Li}_{2}$ identity, or contact me at [email protected] then this could lead to an interesting research contribution. It seems that my methods may be applied to a number of Sun's conjectures. – John M. Campbell Apr 24 '22 at 17:11
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    I sent you a message on ResearchGate. A final remark: all known relations between Li_2 follows (conjecturally) from a five-term functional equation, so one can prove your identity by working on this functional equation hard enough. – pisco Apr 24 '22 at 17:55
  • @pisco Which functional equation is this? – FShrike Apr 24 '22 at 21:26
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    @FShrike "Abel identity" under https://en.wikipedia.org/wiki/Polylogarithm#Dilogarithm – pisco Apr 24 '22 at 21:59
  • @pisco Thank you – FShrike Apr 24 '22 at 22:04

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