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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.00012 |
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Table of Contents:
- We prove the identity \[ 2W_1(x) + \log 4 + ψ\left(\tfrac{1}{2} + x\right) + ψ\left(\tfrac{3}{2} - x\right) = 0, \] where $ψ$ is the digamma function and \[ W_1(x) = 2\int_0^\infty \Re\left( \frac{y}{(y^2+1)(e^{π(y+2ix)} - 1)} \right) dy. \] The identity was first conjectured while studying class number $h(D)$ for $D=m^2$ from two complementary perspectives. Our proof, however, is purely analytic: we compute cosine-series expansions of both sides, expressed in terms of the cosine integral Ci$(z)$. Using the above identity and Möbius inversion we find an elementary formula for $$\sum_{\substack{1\le r<m\\ (r,m)=1}} W_1\!\left(\frac{r}{m}\right).$$