In this work we apply the second Binet formula for Euler’s gamma function Γ ( x ) \Gamma (x) and a Laplace transform formula to derive an infinite series expansion for the auxiliary function f ( x ) f(x) in the computations of sine integral and cosine integral functions in terms of log ⁡ Γ ( x ) \log \Gamma (x) and the Möbius function. Then we apply Möbius inversion to obtain a Kummer type series expansion for log ⁡ Γ ( x ) \log \Gamma (x) . Unlike the original Kummer formula, our formula is not a Fourier series anymore. By differentiating the series expansion for f ( x ) f(x) we obtain an infinite series expansion for the auxiliary function g ( x ) g(x) associated with sine integral and cosine integral functions as well.