A Log-Sine Analogue of Euler's Constant: The Decay Rate of a Sine Product at Primes

For each prime p, we study the product of sin(pi p/k) over the integers k from 2 to p - 1. The logarithm of its absolute value is asymptotic to -C_g p, where C_g = 0.74379486183940054420... . The corresponding exponential base is 0.47530676524152038801... . When the product is restricted to k at most the square root of p, its logarithm is instead asymptotic to minus log(2) times the square root of p. The constant C_g and Euler's constant arise from two averages with respect to the same fractional-part distribution. We relate C_g to the Mellin transforms G and A, derive a geometrically convergent series in even zeta values, and certify the first 100 decimal digits of C_g.