A double parametric integral and a set of positive integers

Calculus Level 5

We define the function \[\phi(a;b):=\int_0^\pi\log(1-2a\cos(bx)+a^2)\,dx.\]

Let \(S\) be the set of positive integers such that \(b\in S\) if and only if \(\phi(a,b)=\phi(0,b)\) for each \(a\in(-1,1)\). If

\[\sum_{s\in S,s\leq 2016}\phi(e^s;1)=k\pi\]

for some integer \(k\), find \(k-924680\).

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