Apery and Fresnel join forces

Calculus Level 5

Let \(\displaystyle S(u) = \int_0^u \sin\left(\frac{\pi}{2} x^2\right) \, dx\) be the Fresnel sine integral. If

\[\sum_{n=1}^{\infty} \frac{S^2(\sqrt{2n})}{n^3}\]

can be expressed in the form \(\dfrac{a}{b}\pi^c\), where \(a\) and \(b\) are coprime positive integers and \(c\) is an integer, find \(a+b+c\).

Hint: Consider an appropriate function \(f(x) = |x|^{\sigma}\) where \(\sigma \in \mathbb{R}\) and apply Parseval's theorem.


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