New bounds and asymptotic expansions for the constants of Landau and Lebesgue

The constants of Landau and Lebesgue are defined, for all integers n⩾0n⩾0, in order, byGn=∑k=0n116k2kk2andLn=12π∫-ππsin(n+12)tsin(12t)dt,which play important roles in the theories of complex analysis and Fourier series, respectively. In this paper, we establish new bounds and asymptotic expansions for the constants of Landau and Lebesgue.