Let $k\ge1$ be an integer and let $\lambda$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $\lambda(n+h_1),\dots, \lambda(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average as well.
Bergeron's conjecture states that, if $1\leq a<b<c<d$ are integers with $ad=bc$, then one has the coefficient-wise inequality ${\binom{b+c}b}_q \ge {\binom{a+d}a}_q$ among two Gaussian polynomials. It originated in algebraic combinatorics and is wide open. The corresponding inequality for binomial coefficients (i.e., t...
In these notes, we introduce the 50-year-old $K(\pi, 1)$ conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in $\mathbb{C}^n$ of a"symmetric"configuration of hyperplanes is a $K(\pi, 1)$ space. Our end goal is to present a proof of the conjecture in the so-called...
Giovanni Paolini· Winter Braids Lecture Notes· 0 citations
Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mathcal{H}$-extensions of $\mathbb{Q}$ ordered by discriminant, where $\mathcal{H}$ is the $3\times 3$ Heisenberg group over $\mathbb{F}_4$. The predicted leading constant...
Let $G=(V,E)$ be a simple graph of order $n$ and let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ be the eigenvalues of its Laplacian matrix. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le |E|+\binom{k+1}{2}$, which was recently confirmed by Kothari and Tudose. Before Brouwer's conjecture w...
We present a short proof of Feige's conjecture: for $n$ independent nonnegative random variables with expectation one, the probability that their sum is less than $n+1$ is at least $\left(\frac{n}{n+1}\right)^n\ge \frac{1}{e}$. The proof was obtained with the assistance of GPT-5.6 Sol and builds on the recent breakthro...
For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conje...
Eric Shen· 0 citations
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