Rivoal's research focuses on several areas of mathematics, including
Diophantine approximation,
Padé approximation, arithmetic
Gevrey series, values of the
Gamma function,
transcendental number theory, and
E-function. His notable contributions include the proof that there is at least one irrational number among nine numbers ζ(5), ζ(7), ζ(9), ζ(11), ..., ζ(21), where ζ is the
Riemann zeta function. Together with
Keith Ball, Rivoal proved that an infinite number of values of ζ at odd integers are linearly independent over , for which he was elected an Honorary Fellow of the
Hardy-
Ramanujan Society. They also proved that there exists an odd number
j such that 1, ζ(3), and ζ(
j) are linear independent over where
2 < j < 170, a specific case of the more general folklore conjecture stating that , ζ(3), ζ(5), ζ(7), ζ(9), ..., are algebraically independent over , which is a consequence of
Grothendieck's period conjecture for mixed Tate
motives. ==See also==