We construct a three-parameter family of rational approximations to values of $q$-hypergeometric series. Using these approximations, we prove that, for every integer $x$ with $|x|\geq2$, the values at $r=x^{-1}$ of Ramanujan's theta function $\psi(r)=\sum_{n\geq0}r^{n(n+1)/2}$, the generating function $\Delta(r)=\sum_{m\geq0}d(2m+1)r^m$ of the divisor function restricted to odd integers, and the generating function $B_4(r)=\sum_{n\geq0}b_4(n)r^n$ for $4$-regular partitions are irrational. We further obtain the upper bounds $18/7$, $18\pi^2/(7\pi^2-24)$, and $3$, respectively, for their irrationality measures. We also show that one of the constructed approximations coincides with the Pad\'e approximation to a Lambert series due to Coussement--Smet.
Let $\beta(s)=\sum_{k=0}^{\infty}(-1)^k(2k+1)^{-s}$ and let $G=\beta(2)$ be Catalan's constant. We develop two families of positive series for reciprocal powers $\beta(s)^{-r}$. The first is obtained from the classical Euler transformation and is evaluated at $1/2$; it is valid for real $s>0$. A second transformation, valid for $s\geq2$, gives a faster series evaluated at $1/3$. We derive finite-sum and integral formulas for the base coefficients, together with a positive recurrence and a composition formula for arbitrary reciprocal powers. When $s$ is an integer, all coefficients are rational. We also give explicit remainder estimates and determine the exact root-convergence rates of the two families. As applications, we obtain positive rational series for every reciprocal power of Catalan's constant and for reciprocal powers of $\pi$ arising from odd values of the Dirichlet beta function.
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fej\'er majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers--Ramanujan and Andrews--Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers--Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.
Let $p$ be a prime number. We prove the sharp Furstenberg set bound in the $p$-adic plane $\mathbb{Q}_p^2$: every $(s,t)$-Furstenberg set $E\subset\mathbb{Q}_p^2$ satisfies $$ \dim_H E\ge \min\left\{s+t,\frac{3s+t}{2},s+1\right\}. $$ This matches the sharp lower bound in the Euclidean plane. We also derive two related consequences: a $p$-adic projection theorem for the maps $\pi_\theta(x,y)=x+\theta y$, together with the corresponding exceptional set estimate giving a $p$-adic analogue of Oberlin's projection question; and a discretized fractal sum-product estimate over $\mathbb{Q}_p$, showing that sufficiently non-concentrated subsets of $\mathbb{Z}_p^\times$ cannot have both small sum set and small product set. The proof follows the projection-theoretic and multiscale machinery developed in the Euclidean works of Orponen-Shmerkin (arXiv:2301.10199) and Ren-Wang (arXiv:2308.08819). The main task is to rebuild this machinery in the non-archimedean setting, and along the way we develop several new $p$-adic inputs needed to overcome the ultrametric features of the problem.
For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schr\"{o}dinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resolved by Bourgain, who constructed counterexamples for the Schr\"{o}dinger maximal operator to show that $s\geq n/(2(n+1))$ is necessary, and Du and Zhang, who proved that $s>n/(2(n+1))$ is sufficient. Rogers, Vargas, and Vega studied the analogous question for the non-elliptic Schr\"{o}dinger maximal operator, where $P = X_1^2-X_2^2 \pm X_3^2\pm \cdots \pm X_n^2$, and proved that, for all $n\geq 2$, $s\geq 1/2$ is necessary and $s>1/2$ is sufficient. In this paper, we construct counterexamples for generalizations of the non-elliptic case and prove a necessary condition of $s\geq 1/2$ for an infinite class of polynomial symbols $P$.
Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. Let $\mathfrak{C}_{\pi}$ be the analytic conductor of $\pi$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\varepsilon>0$ the existence of ineffective constants $c=c_{n,F,\varepsilon}>0$ and $c'=c'_{n,F,\pi',\varepsilon}>0$ such that the standard $L$-function $L(s,\pi)$ satisfies \[ |L(\sigma+it,\pi)|\geq c(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon},\qquad \sigma\geq 1-c(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon} \] and the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$ satisfies \[ |L(\sigma+it,\pi\times\pi')|\geq c'(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon},\qquad \sigma\geq 1-c'(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon}. \] Applications include improvements to the prime number theorems for these $L$-functions and new generalizations of the Brauer-Siegel theorem.
Jesse Thorner· 0 citations
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