On a formula of T. Rivoal

Abstract. In an unpublished 2005 paper T. Rivoal proved the formula $$ \frac{4}{\pi} = \prod_{k \geq 2} \left(1 + \frac{1}{k+1}\right)^{2\rho(k)\lfloor \log_2(k) - 1 \rfloor} $$ where \(\lfloor x \rfloor\) denotes the (lower) integer part of the real number \(x\), and \(\rho(k)\) is the \(4\)-periodic sequence defined by \(\rho(0) = 1\), \(\rho(1) = -1\), \(\rho(2) = \rho(3) = 0\). We show how a lemma in a 1988 paper of J. Shallit and the author allows us to prove that formula, as well as a family of similar formulas involving occurrences of blocks of digits in the base-\(B\) expansion of the integer \(k\), where \(B\) is an integer \(\geq 2\).