Benford's law observes that in many naturally occurring datasets, the leading digit 1 appears about 30% of the time, digit 2 about 17%, and so on, with higher digits progressively rarer. The law follows from the distribution of mantissas: if log₁₀(x) is uniformly distributed modulo 1, then the leading digit d has probability log₁₀(1 + 1/d). This is well understood for sequences like powers of 2, Fibonacci numbers, and many physical measurements.
The joint distribution of leading digits across multiple bases is less understood. What is the joint probability that a number starts with digit 3 in base 10 and digit 5 in base 7? The answer depends on the joint distribution of (log₁₀(x) mod 1, log₇(x) mod 1), which is determined by the arithmetic relationship between log₁₀ and log₇.
The paper connects this to Schanuel's conjecture — one of the major open problems in transcendental number theory. Schanuel's conjecture asserts that if α₁,...,αₙ are Q-linearly independent complex numbers, then the transcendence degree of {α₁,...,αₙ, e^α₁,...,e^αₙ} is at least n. The conjecture implies, among many things, the algebraic independence of logarithms of algebraically independent numbers.
The connection: the joint digit distribution in multiple bases is determined by whether certain logarithms are algebraically independent. If log₁₀(2) and log₇(2) are algebraically independent (as Schanuel's conjecture predicts), then the joint leading-digit distribution is uniform in a precise sense. If they are algebraically dependent (which would refute Schanuel), the joint distribution has correlations.
The digit distribution becomes a litmus test for deep number theory. An empirical observation about leading digits in different bases is equivalent to a statement about the transcendence of logarithmic ratios. The most elementary phenomenon (what digit does a number start with?) bridges to the most abstract conjecture (the algebraic independence of exponentials). The bridge is the logarithm — the same function that explains Benford's law also connects it to transcendental number theory.