An Application of Fourier Series to the Most Significant Digit Problem

It is an old observation [N] that in tables of logarithms, the first few pages are more smudged and worn than the later pages. It follows that the distribution of the first significant digit of the numbers being looked up in the table is not uniform, but skewed toward the smaller digits. In fact, many sets of naturally arising numbers exhibit this asymmetry, with approximately 30% of the numbers beginning with the number 1 and less than 5% beginning with the number 9. Their first significant digit follows a logarithmic distribution, namely, the fraction of numbers whose first significant digit is n is approximately log(n + 1) log n, n = 1, 2, . . ., 9. This phenomenon is known as Benford's Law after Frank Benford, one of the first people to call attention to it. Indeed, he compiled a variety of data totalling 20,299 observations ranging from populations of cities to mathematical tables of powers of integers. The first significant digit in many (but not all) of his data sets were roughly logarithmically distributed. Since Benford's Law is an empirically observed law of nature, rather than a theorem of pure mathematics, there are several competing explanations for it. This phenomenon of leading digits has been extensively investigated and enjoys a colorful history. For an interesting survey of this problem and its literature, see the expository article in the Monthly, August-September, 1976 by Ralph A. Raimi [R]. In all that follows we will write a positive number x in scientific notation, x = mX * lOnX, where 1 < mX < 10 and nX E ;Z. For convenience we will refer to mx as the mantissa of x. Of course, the most significant digit of x is llmx], so it is sufficient to understand the distribution of mx. One explanation of Benford's Law is that it results from "central limit-like" theorems for the mantissas of random variables under multiplicative operations. For example, the figures below show the density functions for the mantissa of the product of 1, 2, 3 and 4 independent and uniformly distributed random variables on [1,10]. The density functions move progressively closer to the log distribution's density function f(t) = 1/(t ln 10). Frequently, naturally occurring data can be thought of as a result of products or quotients of random variables. For example, the population of a city tends to change roughly at a rate proportional to the population. If a city has an initial population P0 and grows by r'% in year i, then the population in n years is Pn = Po(1 + r1)(l + r2) * * * (1 + rn), a product of a number of random variables. In this case, as we'll see, a set of populations of cities would follow Benford's law. More specifically, we'll show for mantissas: