Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function

Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t − 1), where p, q > 0, q = 1 − p, p ≠ q. The variables Mn = ∫01tndL(t), n = 0,1,… are called the moments of the function The principal result of this work is $${M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))$$Mn=nlog2pe−τ(n)(1+O(n−0.99)), where the function τ(x) is periodic in log2x with the period 1 and is given as $$\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}$$τ(x)=121np+Γ'(1)log2p+11n2∂∂zLiz(−qp)|z=1+11n2∑k≠0Γ(zk)Lizk+1(−qp)x−zk, $${z_k} = \frac{{2\pi ik}}{{1n2}}$$zk=2πik1n2, k ≠ 0. The proof is based on poissonization and the Mellin transform.