Ken Ono

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math.NT2026

Ternary polynomial relations and periods of cusp forms

Vasily Ilin, Ken Ono

Bachmann and van Ittersum conjectured that every polynomial $f \in Q [x, y, z]$ satisfying $f (x, y, z) = f (y, z, x), f (x, y, z) + f (- y, x - y, z) + f (y - x, - x, z) = 0$ is a sum of polynom…

math.NT2026

Rogers--Ramanujan identities from the geometry of $X^{a} = Y^{b}$

Yifeng Huang, Kenny Lau, Ken Ono

We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers--Ramanujan and Andrews--Gordon identities for every torus-knot singularity $X…

math.NT2026

Algebraic geometric framework of Rogers--Ramanujan identities

Yifeng Huang, Kenny Lau, Ken Ono +1

The Rogers--Ramanujan identities equate a $q$ -series whose exponents are governed by a quadratic form with an infinite product supported on two residue classes modulo~ $5$ . Identiti…

math.NT2026

Formalized $q$ -series: The Rogers-Ramanujan Identities and Beyond

Kenny Lau, Seewoo Lee, Ken Ono

The theory of $q$ -series and basic hypergeometric series plays a crucial role at the intersection of combinatorics, number theory, and representation theory. From the classical par…

math.NT2026

Modularity of Point Counts for the Curves $X^{a} = Y^{b}$ : New Rogers--Ramanujan Identities

Kenny Lau, Ken Ono

For coprime $1 < a < b$ , let $M_{n}^{a, b} (F_{q})$ be the set of commuting pairs of nilpotent $n \times n$ matrices over $F_{q}$ with $X^{a} = Y^{b}$ . Huang, Jiang, and Oblomkov as…

math.NT2026

Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks

Claudia Alfes, Ken Ono, Ashvin Swaminathan

When $m = 1$ , the Dyson rank generating function is a classical bridge between partition theory, Ramanujan's mock theta functions, and the theory of harmonic Maass forms and nonhol…

math.NT2026

Integer values of $tan (arctan 1 + arctan 2 + \dots + arctan n)$ are rare

Ken Ono

For $n \geq 1$ , we let $x_{n} := tan (k = 1 \sum n arctan k) .$ In 2008, Amdeberhan, Medina, and Moll conjectured that $x_{n} \neq \in Z$ for every $n \geq 5$ . This was kn…

math.NT2026

Thakur's hypotheses on power sums of $F_{q} [t]$

Evan Chen, Ken Ono

In his 2009 paper, Thakur posed three conjectural hypotheses for the degrees of the power sums \[ S_d(k)=\sum_{\substack{a\in \mathbb F_q[t] \text{ monic}\\ °a=d}} a^{-k} \qquad\te…

math.NT2026

Four-digit Kaprekar dynamics in odd bases

Evan Chen, Ken Ono, Richard E. Schwartz +1

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for send…

math.NT2026

ABC implies that Ramanujan's tau function misses almost all primes

David Kurniadi Angdinata, Evan Chen, Chris Cummins +21

Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Raman…

math.NT2026

Almost all primes are partially regular

Evan Chen, Chris Cummins, Ben Eltschig +18

For odd primes $p$ , we let $K_{p} := Q (ζ_{p})$ be the $p$ th cyclotomic field and let $ω$ denote its Teichmuller character. For $α > 1/2$ , we say that an odd prime $p$ is partially…

math.NT2026

Parity of $k$ -differentials in genus zero and one

Dawei Chen, Evan Chen, Kenny Lau +2

Here we completely determine the spin parity of $k$ -differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtaine…

math.NT2025

Parity of the partition function in quadratic progressions

Ken Ono, Ashvin Swaminathan

The parity of the partition function $p (n)$ is one of the most stubborn problems in number theory. In 2010, the first author conjectured, for square-free $1 < D \equiv 23 (mod 24)$ , t…

Ternary polynomial relations and periods of cusp forms

Rogers--Ramanujan identities from the geometry of Xa=Yb

Algebraic geometric framework of Rogers--Ramanujan identities

Formalized q-series: The Rogers-Ramanujan Identities and Beyond

Modularity of Point Counts for the Curves Xa=Yb: New Rogers--Ramanujan Identities

Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks

Integer values of tan(arctan1+arctan2+⋯+arctann) are rare

Thakur's hypotheses on power sums of Fq​[t]

Four-digit Kaprekar dynamics in odd bases

ABC implies that Ramanujan's tau function misses almost all primes

Almost all primes are partially regular

Parity of k-differentials in genus zero and one

Parity of the partition function in quadratic progressions

Rogers--Ramanujan identities from the geometry of $X^{a} = Y^{b}$

Formalized $q$ -series: The Rogers-Ramanujan Identities and Beyond

Modularity of Point Counts for the Curves $X^{a} = Y^{b}$ : New Rogers--Ramanujan Identities

Integer values of $tan (arctan 1 + arctan 2 + \dots + arctan n)$ are rare

Thakur's hypotheses on power sums of $F_{q} [t]$

Parity of $k$ -differentials in genus zero and one