On multiple higher Mahler measures and Witten zeta values associated with semisimple Lie algebras (Q2392991)

The author finds some relations between multiple higher Mahler measures and special values of Witten zeta-functions at positive integers. He proves some general theorem and then, in particular, studies the functions \(f_1(s_1,s_2,s_3)=\sum_{m,n=1}^{\infty} m^{-s_1}n^{-s_2}(m+n)^{-s_3}\) and \(f_2(s_1,s_2,s_3,s_4)=\sum_{m,n=1}^{\infty} m^{-s_1}n^{-s_2}(m+n)^{-s_3}(m+2n)^{-s_4}\). It is shown, for instance, that \[ m(1-xz,1-x,1-z,1-z)=\frac{1}{8}(2f(2,1,1)+f(1,1,2))=\frac{3}{8} \zeta(4)=\frac{\pi^4}{240} \] and \[ m(1-z_1z_2,1-x_2z_1z_2^2,1-x_2,1-z_1,1-z_2)=\frac{1}{16}\Big(3\zeta(5)-\frac{\pi^2}{3}\zeta(3)\Big). \]