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

TL;DR

This paper proves McIntosh's conjecture on Franel integrals involving Bernoulli functions, extends the proof to multiple integrals with various coefficients, and generalizes the results to other Bernoulli functions with odd indices.

Contribution

It provides the first proof of McIntosh's conjecture and introduces new generalizations for integrals involving Bernoulli functions with different parameters and indices.

Findings

Proof of McIntosh's conjecture on Franel integrals.

Extension to integrals with multiple Bernoulli function factors.

Generalization to Bernoulli functions with odd indices.

Abstract

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 periodic Bernoulli function. Secondly, we extend our ideas to prove a similar theorem for $$\int_0^1((a_1x))((a_2x))\cdots ((a_{n}x))\,dx.$$ Lastly, we prove a further generalization in which $((x))$ is replaced by any particular Bernoulli function with odd index.