Jeffrey C. Lagarias

Jeffrey C. Lagarias works in number theory and its connections with computational complexity theory, geometry and dynamical systems. One striking contribution is a reformulation of the Riemann hypothesis using only harmonic numbers, integer divisors, the exponential function and the natural logarithm. ^[ref][ref]

Write $H_n={\sum }_{j=1}^{n}1/j$ and let $\sigma (n)$ be the sum-of-divisors function. Lagarias proved that the Riemann hypothesis is equivalent to the assertion that, for every positive integer $n$,with equality only at $n=1$.^[ref]

For example, $\sigma (6)=1+2+3+6=12$ and $H_6=49/20$, so the right side is approximately $12.83$. This verifies one instance. The force of the theorem is the universal quantifier: checking any finite collection of integers does not establish the entire inequality. An elementary-looking statement can therefore encode a deep question about the zeros of a function of the Riemann zeta function.