criterion, the left hand side is always 1, whereas the right hand side is zero, because the sequence is countable, so f is zero almost everywhere. In fact, the de Bruijn–Post Theorem states the converse of the above criterion: If f is a function such that the criterion above holds for any equidistributed sequence in [a, b], then f is Riemann-integrable in [a, b]. Equidistribution modulo 1 A sequence (a1, a2, a3, ...) of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by (an) or by an − ⌊an⌋, is equidistributed in the interval [0, 1]. Examples The equidistribution theorem: The sequence of all multiples of an irrational α, 0, α, 2α, 3α, 4α, ... is equidistributed modulo 1. More generally, if p is a polynomial with at least one coefficient other than the constant term irrational then the sequence p(n) is uniformly distributed modulo 1. This was proven by Weyl and is an application of van der Corput's difference theorem. The sequence log(n) is not uniformly distributed modulo 1. This fact is related to Benford's law. The sequence of all multiples of an irrational α by successive prime numbers, 2α, 3α, 5α, 7α, 11α, ... is equidistributed modulo 1. This is a famous theorem of analytic number theory, published by I. M. Vinogradov in 1948. The van der Corput sequence is equidistributed. Weyl's criterion Weyl's criterion states that the sequence an is equidistributed modulo 1 if and only if for all non-zero integers ℓ, The criterion is named after, and was first formulated by, Hermann Weyl. It allows equidistribution questions to be reduced to bounds on exponential sums, a fundamental and general method. {| class="toccolours collapsible collapsed" width="90%" style="text-align:left" !Sketch of proof |- |If the sequence is equidistributed modulo 1, then we can apply the Riemann integral criterion (described above) on the function which has integral zero on the interval [0, 1]. This gives Weyl's criterion immediately. Conversely, suppose Weyl's criterion holds. Then the Riemann integral criterion holds for functions f as above, and by linearity of the criterion, it holds for f being any trigonometric polynomial. By the Stone–Weierstrass theorem and an approximation argument, this extends to any continuous function f. Finally, let f be the indicator function of an interval. It is possible to bound f from above and below by