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 breakthrough of Vlassis and Thomas establishing Gaffke's conjecture on the finite-sample validity of a distribution-free $p$-value. We also discuss the implications of the subsequent work of Ming, Ramdas, Shen, Wang, and Waudby-Smith.
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 is not a single Euler product, but rather a sum of two distinct Euler products. Our methods also give an efficient algorithm for computing the conjectural Loughran-Santens leading constant for many $2$-groups of nilpotency class $2$.
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., the case $q=1$) must be known to experts, but we could not find it in the literature. We give two proofs, each generalizing the statement in a separate direction. Binomial coefficients (and Gaussian polynomials) are fundamental combinatorial objects, and so one naturally hopes to see a combinatorial proof of this inequality. However, this seems hard to come by. We nevertheless give a combinatorial proof of a special case.
Determining the properties of a random polynomial has fuelled significant investigation over the past century. One driving force of this research is a 1936 paper of Van der Waerden. Fix $n \geq 3$ and let $E_n(B)$ be the number of monic, irreducible, non-$S_n$ polynomials $f = X^n + a_1 X^{n-1} + \cdots + a_n$ with $|a_j| \leq B$ for all $j$. A recent breakthrough of Bhargava bounds $E_n(B) \ll B^{n-1}$. This spectacularly resolves a conjecture of Van der Waerden, but leaves open its stronger form, namely that $E_n(B) = o(B^{n-1})$. Inspired by recent progress, we now address this strong form. Bhargava's result, together with work of Chow and Dietmann, essentially reduces the strong Van der Waerden conjecture to the claim that the number of polynomials $f$ with Galois group $A_n$ is $o(B^{n-1})$. Assuming a twisted function field version of Chowla's conjecture, we prove this claim. This not only connects two active and challenging areas of research, but also conditionally resolves the strong Van der Waerden conjecture for all $n \geq 7$. Our proof is based on a variant of Heath-Brown and Pierce's square sieve and $q$-van der Corput differencing. Our methods also apply to the analogous problem of counting square discriminants of polynomials that are not necessarily monic. In addition to describing our new contributions, we briefly elaborate on the various conjectures appearing in Van der Waerden's paper and some of the exciting recent work of others in this area of arithmetic statistics.
Theresa C. Anderson, Sam Chow, R. Dietmann et al.· 0 citations
In this paper, we prove that the set of counterexamples to uniform Littlewood's conjecture proposed in \cite{BFK25}, that is, the set of real pairs $(x,y)$ satisfying $$\limsup_{Q\to+\infty}\ Q\min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0$$ is hyperplane absolute winning. In particular, it has full Hausdorff dimension in $\mathbb{R}^2$.
Gilbreath's conjecture asserts that if one starts with the sequence of primes and takes successive absolute differences to create a triangular array, then the left diagonal of this array consists entirely of ones after the first row. In this paper, we show that the analogue of this conjecture for a Cram\'er random model holds, in which the (normalized) prime gaps are replaced by independent random variables with geometric distributions of logarithmic size. We also give some preliminary analysis of the associated continuous probabilistic model for this problem, as well as a deterministic"inverse theorem"that isolates the specific obstructions to Gilbreath's conjecture (assuming a Cram\'er type bound on prime gaps), namely long blocks of zeroes, or very long shallow $\{0,d\}$-valued blocks for some $d \geq 2$.
Zachary A. Chase, Zach Hunter, Terence Tao· 2 citations
In this paper, we study a Riesz-type weighted sum of the Liouville function, $f(x):=-\sum_{n\le x}\frac{\lambda(n)\log n}{\sqrt n}\log\frac{x}{n}$ for sufficiently large $x$. Motivated by a recent research on the sign criteria for the Riemann Hypothesis arising from weighted prime-counting functions, we investigate the sign behavior of $f(x)$ and its relation to the zeros of the Riemann zeta function. We first prove that if $f(x)$ is non-negative for all sufficiently large $x$, then the Riemann Hypothesis holds. The proof is based on the Mellin transform of $f$ and the analytic properties of $\frac{\zeta(2s)}{\zeta(s)}$. Conversely, assuming the Riemann Hypothesis, the Simple Zero Conjecture, and an absolute convergence condition involving the nontrivial zeta zeros, we derive an explicit formula for $f(x)$ in terms of these zeros. In particular, we show that $f(x)\sim \frac{(\log x)^3}{12|\zeta(\frac{1}{2})|}$ as $x\to \infty$. Consequently, we show that under these hypotheses, $f(x)$ is eventually positive.
