  We find closed-form expressions for the Schur indices of 4d
$\mathcal{N}=2^{*}$ super Yang-Mills theory with unitary gauge groups for
arbitrary ranks via the Fermi-gas formulation. They can be written as a sum
over the Young diagrams associated with spectral zeta functions of an ideal
Fermi-gas system. These functions are expressed in terms of the twisted
Weierstrass functions, generating functions for quasi-Jacobi forms. The indices
lie in the polynomial ring generated by the Kronecker theta function and the
Weierstrass functions which contains the polynomial ring of the quasi-Jacobi
forms. The grand canonical ensemble allows for another simple exact form of the
indices as infinite series. In addition, we find that the unflavored Schur
indices and their limits can be expressed in terms of several generating
functions for combinatorial objects, including sum of triangular numbers,
generalized sums of divisors and overpartitions.
