Mathematical Proofs and Problems Captivating Researchers in 2026

Iggidr (2026). Character sums are central tools in analytic number theory, controlling the distribution of primes in arithmetic progressions. This paper establishes sharp bounds on the frequency with which mixed character sums achieve unusually large values — results with implications for prime gaps, zero-free regions, and the Generalised Riemann Hypothesis.

Fouvry, Kowalski, Michel & Sawin (2026). The second paper in a series developing the theory of toroidal families of automorphic forms as a tool for computing averages and moments of L-functions. Cubic moments are significantly harder to handle than second moments and require deep tools from the Langlands programme — this represents the current frontier of analytic number theory.

Anttila, Fraser & Koivusalo (2026). Studies the metric Diophantine approximation properties of fractal sets with non-linear structure, characterising which points on certain self-affine carpets fail to be well-approximated by rationals. Connects the classical theory of continued fractions to modern fractal geometry through Hausdorff dimension calculations.

Sehrawat (2026). A number-theoretic study motivated by post-quantum cryptography: the security of the PRIM-LWE lattice problem depends on properties of primitive roots modulo primes. This paper computes the density of certain determinant conditions involving primitive roots, providing concrete hardness evidence for a promising post-quantum candidate.

Mihara (2026). Extends the classical statistical technique of principal component analysis to p-adic number fields — a fascinating intersection of machine learning and non-Archimedean mathematics. Beyond its intrinsic mathematical interest, p-adic PCA has potential applications in analysing data with ultrametric structure, including phylogenetic trees and certain network hierarchies.

Chen (2026). Arthur packets are fundamental objects in the representation theory of reductive groups, organising automorphic forms into families with shared L-functions. This paper constructs local Arthur packets for metaplectic groups — the non-algebraic double covers of symplectic groups — and verifies the Adams conjecture in this setting.

Yu (2026). A careful expository treatment of Dieudonne modules — the p-adic linear algebra objects that classify abelian varieties in characteristic p — with new perspectives that make the theory more accessible to researchers approaching it from arithmetic geometry rather than formal group theory. An important pedagogical contribution to a technically demanding field.

Jakhar & Ray (2026). Investigates the distribution of lattice shapes — the geometric invariants of unit groups — of multiquadratic number fields, extending classical results about quadratic fields to compositum towers. The limiting distribution of shapes connects to deep questions in the geometry of numbers and the distribution of abelian extensions.

Cheng & Zhang (2026). A focused study of the exact 2-adic valuation of the sum-of-k-th-powers-of-divisors function sigma_k(n) — a classical arithmetic function whose p-adic properties control important aspects of modular form congruences. The results provide new explicit formulas that complement existing results for odd primes.

Iggidr (2026). Character sums are central tools in analytic number theory, controlling the distribution of primes in arithmetic progressions. This paper establishes sharp bounds on the frequency with which mixed character sums achieve unusually large values — results with implications for prime gaps, zero-free regions, and the Generalised Riemann Hypothesis.

Fouvry, Kowalski, Michel & Sawin (2026). The second paper in a series developing the theory of toroidal families of automorphic forms as a tool for computing averages and moments of L-functions. Cubic moments are significantly harder to handle than second moments and require deep tools from the Langlands programme — this represents the current frontier of analytic number theory.

Anttila, Fraser & Koivusalo (2026). Studies the metric Diophantine approximation properties of fractal sets with non-linear structure, characterising which points on certain self-affine carpets fail to be well-approximated by rationals. Connects the classical theory of continued fractions to modern fractal geometry through Hausdorff dimension calculations.

Sehrawat (2026). A number-theoretic study motivated by post-quantum cryptography: the security of the PRIM-LWE lattice problem depends on properties of primitive roots modulo primes. This paper computes the density of certain determinant conditions involving primitive roots, providing concrete hardness evidence for a promising post-quantum candidate.

Mihara (2026). Extends the classical statistical technique of principal component analysis to p-adic number fields — a fascinating intersection of machine learning and non-Archimedean mathematics. Beyond its intrinsic mathematical interest, p-adic PCA has potential applications in analysing data with ultrametric structure, including phylogenetic trees and certain network hierarchies.

Chen (2026). Arthur packets are fundamental objects in the representation theory of reductive groups, organising automorphic forms into families with shared L-functions. This paper constructs local Arthur packets for metaplectic groups — the non-algebraic double covers of symplectic groups — and verifies the Adams conjecture in this setting.

Yu (2026). A careful expository treatment of Dieudonne modules — the p-adic linear algebra objects that classify abelian varieties in characteristic p — with new perspectives that make the theory more accessible to researchers approaching it from arithmetic geometry rather than formal group theory. An important pedagogical contribution to a technically demanding field.

Jakhar & Ray (2026). Investigates the distribution of lattice shapes — the geometric invariants of unit groups — of multiquadratic number fields, extending classical results about quadratic fields to compositum towers. The limiting distribution of shapes connects to deep questions in the geometry of numbers and the distribution of abelian extensions.

Cheng & Zhang (2026). A focused study of the exact 2-adic valuation of the sum-of-k-th-powers-of-divisors function sigma_k(n) — a classical arithmetic function whose p-adic properties control important aspects of modular form congruences. The results provide new explicit formulas that complement existing results for odd primes.