# Euler-Mascheroni Mashup

Calculus Level 5

Suppose the sum $\sum_{n=1}^\infty \left[ H_n - \gamma - \ln n - \dfrac{\zeta(2n)}{2n} \right]$ can be expressed in the form $\frac{1}{m} \big(a + b\gamma - c\ln(k\pi) \big),$ where $$a, b, c,$$ and $$m$$ are positive integers and $$\gcd(a,b,c,m) = 1$$.

Find $$a+b+c+k+m$$.

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