A half-shift reflection identity for the digamma function
Nikita Kalinin
TL;DR
The article establishes a pure analytic half-shift reflection identity for the digamma function by proving $2W_1(x) + \log 4 + \psi\left(\tfrac{1}{2}+x\right) + \psi\left(\tfrac{3}{2}-x\right) = 0$, where $W_1(x)$ is given by a specific integral. The core method expands both sides into cosine Fourier series, expressing the digamma sum in terms of the cosine integral $\mathrm{Ci}$ and $W_1(x)$ via Laplace–Euler integrals, enabling term-by-term comparison. This yields the explicit cosine-coefficient identities $a_k = 4(-1)^k \mathrm{Ci}(\pi k)$ for $\psi(\tfrac{1}{2}+x)+\psi(\tfrac{3}{2}-x)$ and $a_k = 2(-1)^{k+1} \mathrm{Ci}(\pi k)$ for $W_1(x)$, culminating in the main equality. As a byproduct, the authors derive an explicit formula for the reduced-residue sum $\sum_{\substack{1\le r<m\,(r,m)=1}} W_1\left(\tfrac{r}{m}\right)$ in terms of Euler’s totient, primes dividing $m$, and $\psi$, enriching connections to class-number formulas and topographs.
Abstract
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 $$\sum_{\substack{1\le r<m\\ (r,m)=1}} W_1\!\left(\frac{r}{m}\right).$$