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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