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By Jeffrey C. Lagarias. American Mathematical Monthly 109(6), 534–543 (2002). DOI doi.org/10.1080/00029890.2002.11919883 .

Author-hosted version dated 29 July 2001: websites.umich.edu/~lagarias/doc/elementaryrh.pdf

Verified locator: first page, Problem E, equation (1.1) and Theorem 1.1; harmonic numbers and the sum-of-divisors inequality equivalent to the Riemann hypothesis.

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.

Nickolas Andersen studies analytic number theory, modular forms and L-functions. With Jeffrey C. Lagarias and Robert C. Rhoades, he studied shifted polyharmonic Maass forms, which replace an eigenfunction equation byHere ${\Delta }_k$ is a weight-dependent hyperbolic Laplacian; the functions also obey arithmetic symmetry and controlled growth. The study proves finite dimensionality and describes the role of Eisenstein series and their parameter derivatives. ^[ref][ref]

The mechanism is visible by differentiating a family of eigenfunctions. If $LE(s)=\lambda (s)E(s)$ and $L$ is independent of $s$, thenAt a fixed parameter value, applying $L-\lambda$ again gives zero. A derivative can therefore yield a generalized solution of depth two instead of another ordinary eigenfunction. The arithmetic symmetry and growth requirements still need checking; the operator identity explains the construction's motivation.

By Jeffrey C. Lagarias and Robert C. Rhoades. Ramanujan Journal 41, 191–232 (2016), DOI doi.org/10.1007/s11139-015-9729-1 . The preprint was submitted in 2015.

Verified locators: preprint pages 1–2, equations (1.1)–(1.2), modularity, growth condition and iterated Laplacian; pages 3–4, Theorem 1.1 and construction of a Taylor basis from Eisenstein series.

Robert C. Rhoades coauthored work extending the study of Maass forms beyond a single eigenfunction equation. With Jeffrey C. Lagarias, he studied polyharmonic Maass forms: functions with the required arithmetic symmetry and growth conditions that are annihilated by a power of a weight-dependent hyperbolic Laplacian, ${\Delta }_k^{m}f=0$. Their paper constructs bases using derivatives of Eisenstein series with respect to a parameter.^[ref]

An elementary analogy explains why taking a power changes the problem. For the derivative operator $D=d/dt$, the function $f(t)=t$ satisfies $D^{2}f=0$ but $Df=1\ne 0$. Thus requiring two applications to give zero allows more functions than requiring one. The arithmetic setting adds essential symmetry and growth restrictions; the analogy alone does not produce a Maass form. The paper shows how these restrictions lead to explicit finite-dimensional spaces, rather than the unrestricted solution sets of a differential equation.^[ref]

By Nickolas Andersen, Jeffrey C. Lagarias and Robert C. Rhoades.

Verified locator: Page 1, equation (1.1) and footnotes 1–2: classical Maass forms, moderate growth, sign conventions.