# 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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