487 papers in the wider index

Number theory

Primes, modular forms, and arithmetic structure.

Integer sequences Graph theory Topology Probability Geometry Analysis Logic

In this topic5 papers in the archive

Preprint · Oct 2026

Searching for a Decreasing Function in the Collatz Conjecture

This paper investigates whether a Lyapunov-style function can be found that decreases along Collatz trajectories, and concludes that no such function is established here: the work is a structured research programme, not a proof. Starting from the failed candidate F(n) = n, the logarithmic potential F(n) = log₂ n reveals an average downward drift once the process is compressed to odd integers; residue classes modulo powers of 2 then explain how many divisions by 2 follow each odd step, and a residue-dependent correction on classes modulo a power of 2 turns part of the problem into an optimisation problem. The aim is to show how an elementary-looking problem becomes a serious investigation spanning dynamical systems, probability, number theory, optimisation and computation.

Taha Tanver Athar, Tanver Athar18 reads

Preprint · Jul 2026

The Echo-Square Coprime sequence: a new integer sequence with index-dependent arithmetic and digit-sum constraints

We introduce the Echo-Square Coprime (ESC) sequence, a greedy smallest-unused-integer construction binding each term to its own index through three simultaneous constraints: coprimality with n, a triangular digit sum, and a square echo requiring a(n) to be a perfect square exactly when n is. The conjunction produces sharp emergent spikes at every square index, and the sequence is provably not a permutation of the positive integers.

Taha Tanver Athar, Tanver Athar42 reads

arXiv:2511.0442

A spectral proof of the Riemann hypothesis

We construct a self-adjoint operator whose eigenvalues coincide with the nontrivial zeros of the Riemann zeta function, yielding a conditional path to the long-standing conjecture via spectral theory.

D. Whitfield, L. Moreau9,402 reads

arXiv:2409.7745

Convergence rates in the Collatz conjecture

New asymptotic bounds on stopping times for the 3x+1 iteration, obtained via a stochastic model refined with recent lower bounds on nontrivial cycles.

S. Terrance982 reads

arXiv:2501.0987

O-minimal structures and unlikely intersections

We prove a uniform bound on the complexity of intersections of definable families with algebraic subvarieties, giving a new route to unlikely-intersection statements in the spirit of the Zilber–Pink conjecture.

R. Kaniadakis, T. Zhou1,521 reads