Dirichlet character sums for the terms of the von Mangoldt function

\sum _{n=1}^{\infty } (1 \chi _{1,1}(n)+0)

\log(2) = \sum _{n=1}^{\infty } (2 \chi _{2,1}(n)-1)

\log(3) = \sum _{n=1}^{\infty } (3 \chi _{3,1}(n)-2)

\log(2) = \sum _{n=1}^{\infty } (2 \chi _{4,1}(n)-1)

\log(5) = \sum _{n=1}^{\infty } (5 \chi _{5,1}(n)-4)

\log(1) = \sum _{n=1}^{\infty } (2 \chi _{2,1}(n)-1) (3 \chi _{3,1}(n)-2)

\log(7) = \sum _{n=1}^{\infty } (7 \chi _{7,1}(n)-6)

\log(2) = \sum _{n=1}^{\infty } (2 \chi _{8,1}(n)-1)

\log(3) = \sum _{n=1}^{\infty } (3 \chi _{9,1}(n)-2)

\log(1) = \sum _{n=1}^{\infty } (2 \chi _{2,1}(n)-1) (5 \chi _{5,1}(n)-4)

\log(11) = \sum _{n=1}^{\infty } (11 \chi _{11,1}(n)-10)

\log(1) = \sum _{n=1}^{\infty } (4 \chi _{4,1}(n)-1) (3 \chi _{3,1}(n)-2)

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