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,...
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...
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...
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...
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 resol...
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 $\varepsi...
Jesse Thorner· 0 citations
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