Proofs of McIntosh's Conjecture on Franel Integrals and Two Generalizations

Bruce C. Berndt, Likun Xie, Alexandru Zaharescu

We provide a proof of a conjecture made by Richard McIntosh in 1996 on the values of the Franel integrals, $${\int }_0^1((ax))((bx))((cx))((ex))dx,$$ where $((x))$ is the first peri…