In this paper, we prove an exceptional sign pair phenomenon for three $q$-series from Ramanujan's Lost Notebook, namely $v_2(q),v_3(q)$, and $v_4(q)$. This was first observed by Andrews in a 1986 paper. Building on the recent work of Kundu, Storzer, Wang, and the author, which established that the coefficients of these $q$-series are alternating in sign except in a density-zero set, we study the exceptional indices where the alternating sign pattern fails. We prove that these exceptional indices occur infinitely often in a structured manner. More precisely, we show that there exist infinitely many pairs of consecutive coefficients having the same sign, and that at least one coefficient in each pair is a local minimum of the sequence of absolute values of the coefficients. Our proof combines precise asymptotic expansions for the coefficients with a careful analysis of an oscillatory factor appearing in the expansions which governs the exceptional sign behavior.
We prove the strict positivity conjectured by Chenevier for the critical vectors in the unconditional part of his automorphic Hermite--Minkowski theorem. The proof establishes strict negativity of all Verblunsky coefficients of a circle measure associated with the weight $(\arcsin x)/x$ on $(-1,1)$. A positive-kernel f...
In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called N\"orlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the...
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density a...
In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit...
We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over $\mathbb{Q}$ with non-constant $j$-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main $p^2$ term, a $p^{3/2}$ term arising from the first c...
Lucas Chen, Joshua Im, Steven J. Miller et al.· 0 citations
In 1965, Chowla conjectured that the signs of the Liouville function become asymptotically uncorrelated at any fixed collection of distinct shifts. In 2015, Matom\"aki, Radziwi\l\l{} and Tao proved an averaged form of Chowla's conjecture. In 2022, Lichtman proved a variant of this conjecture over primes on average. In...
Biao Wang· 0 citations
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