bibkey: nist2026lerch authors: NIST Digital Library of Mathematical Functions year: 2026 title: Hurwitz Zeta Function doi: null url: https://dlmf.nist.gov/25.11.E18 claim: The derivative of the Hurwitz zeta function at zero equals log Gamma of its parameter minus one half of log two pi. strata_touched:

• D5/S3/Analytic/HolonomyDeterminant/ReflectedHurwitzDerivative license: citation-only triage: anchor

Lerch’s derivative formula

Equation (25.11.18) gives ζ′(0,a) = log Γ(a) − log(2π)/2. For a real parameter strictly between zero and one, the same identity at 1-a gives the reflected formula. Reflection on the additive circle sends the class of a to the class of 1-a.

The Lean proof uses regularized differences of Hurwitz and Riemann finite sums. Their increments are interpolation remainders with a summable uniform bound. The limit is identified by analytic continuation; convergence of its derivatives and the Bohr-Mollerup limit for log Gamma determine the derivative at zero. The constant is supplied by the Riemann zeta derivative at zero.

Verified locator

• URL: https://dlmf.nist.gov/25.11.E18

The equation’s TeX representation at https://dlmf.nist.gov/25.11.E18.tex states the displayed identity with the derivative in the first zeta argument.