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We develop a new theory of $L$-series based on replacing Dirichlet charactersmod $N$ by symmetric functions mod $N$ in order to calculate explicitly thesums of infinite series more closely related to $\zeta(2n+1)$, specifically$\zeta(3)$, $\zeta(5)$. This generalizes the corresponding theory of sums of$L$-series associated to Dirichlet characters.

$L$-series associated to symmetric functions mod $N$ with applications related to $\zeta(3)$, $\zeta(5)$