Asymptotic expansion for the Lebesgue constants of the Walsh system

Let Lk denote the Lebesgue constants of the Walsh system. The following exact result is established by means of Mellin transforms: ∑1≤k<n Lk=n4log2n+nF(log2n)−Ln2 forn⩾1 where F(u) is a continuous periodic function with period 1 whose Fourier coefficients can be explicitly expressed in terms of Riemannʹs zeta function. This improves an old result of Fine.