arXiv subjects

Vedant Rathi

Publications and source records attributed to Vedant Rathi.

6 recordsLinked to original sources

arXiv2026

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

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 Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |\tau(n)|=\ell \text{ for some } n\ge 1\} = O(X^{13/22}), \] which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that $S(X)$ should nevertheless be infinite, with predicted order of magnitude \[ S(X)\asymp \frac{C X^{\frac{1}{11}}}{(\log X)^2}. \] The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.

math.NT↗

arXiv2026

Almost all primes are partially regular

For odd primes $p$, we let $K_p:=\mathbb{Q}(\zeta_p)$ be the $p$th cyclotomic field and let $\omega$ denote its Teichmuller character. For $\alpha>1/2$, we say that an odd prime $p$ is partially regular if the eigenspaces of the $p$-Sylow subgroup of $\operatorname{Cl}(K_p)$ under the Galois action vanish for all characters $\omega^{p-2k}$ with \[ 2\le 2k \le \frac{\sqrt{p}}{(\log p)^{\alpha}}. \] Equivalently, $p\nmid \operatorname{num}(B_{2k})$ throughout this range. We prove that a density-one subset of primes is partially regular in this sense. By Leopoldt reflection, this yields a partial Vandiver Theorem: for a density-one set of primes $p$, the even eigenspaces $A_p(\omega^{2k})$ vanish for all even $2k$ satisfying the inequality above. This result has consequences for Kubota-Leopoldt $p$-adic $L$-functions, congruences between cusp forms and Eisenstein series, and $p$-torsion in algebraic $K$-groups. The theorem proving partial regularity for almost all $p$ is fully formalized in Lean/Mathlib and was produced automatically by AxiomProver from a natural-language statement of the conjecture.

math.NT↗

arXiv2026

Dead ends in square-free digit walks

We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near $1$, corresponding to an asymptotic dead-end density of $\approx 5.218\times 10^{-5}$. We prove that the true asymptotic dead-end density satisfies \[ c_{\mathrm{dead}} \approx 1.317\times 10^{-9}, \] roughly a factor of $\sim 4\times 10^4$ smaller than the prediction. For every base $b\geq 2$, we prove that dead-end densities exist and are given by a closed-form expression (as a finite alternating sum of Euler products). The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the problem.

math.CO↗

arXiv2026

Fel's Conjecture on Syzygies of Numerical Semigroups

Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $\sigma_k=\sum_i d_i^k$ (and $\delta_k=(\sigma_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture.

math.CO↗

arXiv2025

The Challenge of Teaching Reasoning to LLMs Without RL or Distillation

Reasoning-capable language models achieve state-of-the-art performance in diverse complex tasks by generating long, explicit Chain-of-Thought (CoT) traces. While recent works show that base models can acquire such reasoning traces via reinforcement learning or distillation from stronger models like DeepSeek-R1, previous works demonstrate that even short CoT prompting without fine-tuning is able to improve reasoning. We ask whether long CoT can be induced in a base model using only prompting or minimal tuning. Using just 20 long CoT examples from the reasoning model \texttt{QwQ-32B-Preview}, we lightly fine-tune the base model \texttt{Qwen2.5-32B}. The resulting model outperforms the much larger \texttt{Qwen2.5-Math-72B-Instruct}, showing that a handful of high-quality examples can unlock strong reasoning capabilities. We further explore using CoT data from non-reasoning models and human annotators, enhanced with prompt engineering, multi-pass editing, and structural guidance. However, neither matches the performance of reasoning model traces, suggesting that certain latent qualities of expert CoT are difficult to replicate. We analyze key properties of reasoning data, such as problem difficulty, diversity, and answer length, that influence reasoning distillation. While challenges remain, we are optimistic that carefully curated human-written CoT, even in small quantities, can activate reasoning behaviors in base models. We release our human-authored dataset across refinement stages and invite further investigation into what makes small-scale reasoning supervision so effective.

cs.AI↗

arXiv2025

Breaking Bad Tokens: Detoxification of LLMs Using Sparse Autoencoders

Large language models (LLMs) are now ubiquitous in user-facing applications, yet they still generate undesirable toxic outputs, including profanity, vulgarity, and derogatory remarks. Although numerous detoxification methods exist, most apply broad, surface-level fixes and can therefore easily be circumvented by jailbreak attacks. In this paper we leverage sparse autoencoders (SAEs) to identify toxicity-related directions in the residual stream of models and perform targeted activation steering using the corresponding decoder vectors. We introduce three tiers of steering aggressiveness and evaluate them on GPT-2 Small and Gemma-2-2B, revealing trade-offs between toxicity reduction and language fluency. At stronger steering strengths, these causal interventions surpass competitive baselines in reducing toxicity by up to 20%, though fluency can degrade noticeably on GPT-2 Small depending on the aggressiveness. Crucially, standard NLP benchmark scores upon steering remain stable, indicating that the model's knowledge and general abilities are preserved. We further show that feature-splitting in wider SAEs hampers safety interventions, underscoring the importance of disentangled feature learning. Our findings highlight both the promise and the current limitations of SAE-based causal interventions for LLM detoxification, further suggesting practical guidelines for safer language-model deployment.

cs.CL↗