Explicit Bounds for the Approximation Er- Ror in Benford’s Law

Abstract Benford’s law states that for many random variables X > 0 its leading digit D = D(X) satisfies approximately the equation P(D = d) = log10(1 + 1/d) for d = 1, 2, . . . , 9. This phenomenon follows from another, maybe more intuitive fact, applied to Y := log10 X: For many real random variables Y , the remainder U := Y − ⌊Y ⌋ is approximately uniformly distributed on [0, 1). The present paper provides new explicit bounds for the latter approximation in terms of the total variation of the density of Y or some derivative of it. These bounds are an interesting and powerful alternative to Fourier methods. As a by-product we obtain explicit bounds for the approximation error in Benford’s law.