= Jeffrey C. Lagarias
{c}
<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>. https://websites.umich.edu/~lagarias/[Author's research interests]{ref}<An Elementary Problem Equivalent to the Riemann Hypothesis by Jeffrey C. Lagarias>[Problem E and Theorem 1.1]{ref}
Write $H_n=\sum_{j=1}^n1/j$ 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> $n$,
$$
\sigma(n)\leq H_n+e^{H_n}\log H_n,
$$
with equality only at $n=1$.<An Elementary Problem Equivalent to the Riemann Hypothesis by Jeffrey C. Lagarias>[equation (1.1) and Theorem 1.1]{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>.
\Image[https://websites.umich.edu/~lagarias/lagarias-10.jpg]
{title=Jeffrey C. Lagarias}
{description=Portrait identified on https://websites.umich.edu/~lagarias/[the author's University of Michigan homepage].}
{height=300}
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