Benford's Law: Why the IRS might care about the 3x+1 problem and ζ(s)

Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers or of $2^n$ equals 1 about 30% of the time; the IRS uses this digit bias to detect fraudulent corporate tax returns. This phenomenon, known as Benford's Law, was first noticed by observing which pages of log tables were most worn from age -- it's a good thing there were no calculators 100 years ago! We'll discuss the general theory and application, talk about some fun examples (ranging from the $3x+1$ problem to the Riemann zeta function to fragmentation problems, as time permits), and see how the irrationality type of numbers often enter into the analysis (through error terms in equidistribution theorems).