Proofs of McIntosh's conjecture on Franel integrals and two generalizations (Q6042807)

\textit{J. Franel} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1924, 198--201 (1924; JFM 50.0119.01)] gave the formula \[ I(a,b) := \int_0^1 ((ax))((bx))\,dx = \frac{(a,b)^2}{12ab}, \] where \(a\) and \(b\) are positive integers, \((a, b) := \operatorname{gcd}(a,b)\), and \(((x)) := x - \lfloor{x}\rfloor - 1/2\) is the sawtooth function; here \(\lfloor{x}\rfloor\) denotes the greatest integer~\(\le x\). By the integrand's asymmetry about the midpoint \(x = 1/2\), it is easy to see that the corresponding integral of the product \(((ax))((bx))((cx))\) is \(I(a,b,c) = 0\). Then, the interest is to study \[ I(a,b,c,e) := \int_0^1 ((ax))((bx))((cx))((ex))\,dx, \] where \(a, b, c\) and \(e\) are positive integers, or, with greater generality, \[ I(a_1,a_2,\dots,a_n) := \int_0^1 ((a_1x))((a_2x)) \cdots ((a_nx))\,dx, \] where \(n\) is any even, positive integer. In 1996, McIntosh stated some properties for \(I(a,b,c,e)\), and conjectured that \[ f(a, b, c, e) := \frac{240 a^3 b^3 c^3 e^3 (a,b,c) (a,b,e) (a,c,e) (b,c,e)} {(a,b)^2 (a,c)^2 (a,e)^2 (b,c)^2 (b,e)^2 (c,e)^2 (a,b,c,e)^4} I(a,b,c,e) \] is an integer for any positive integers \(a, b, c, e\). The paper under review proves McIntosh's conjecture, and states a similar theorem for \(I(a_1,a_2,\dots,a_n)\). Moreover, \(((x)) = B_1(x-\lfloor{x}\rfloor)\) where \(B_1(x) = x-1/2\) is the Bernoulli polynomial of degree \(1\) (i.e., \(((x))\) is the first Bernoulli periodic function). Then, the paper gives a further generalization in which \(((x))\) is replaced by the periodic function \(B_k(x-\lfloor{x}\rfloor)\) with \(B_k(x)\) any Bernoulli polynomial of odd degree.