Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that \[ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha}<1. \] By his variation formula, this inequality is sufficient for every positive zero of $\pi_n$ to move to the right as $c$ increases. For $0<c<1$, a first-crossing argument proves the conjecture throughout $0<\alpha<\beta$. Together with the earlier regions recorded by Gautschi and established by Milovanovi\'c, this settles $\beta\geq0$. In the negative wedge, writing $\alpha=-r-\lambda$ and $\beta=-r+\lambda$, an ensemble Ward bound yields the region $c^2\leq3/(3+r)$. A strengthening of the same crossing lemma, using an orthogonal expansion and Markov's theorem, removes this restriction. Consequently, the conjecture holds for every $n\geq1$, $-1<\alpha<\beta$, and $0<c\leq1$. We also give a direct degree-one proof and an explicit asymptotic limit. The case $c=1$ is immediate.
For $-1<\alpha\leq1/2$, let $J_\alpha$ be the Bessel function of the first kind and let $j_{\alpha,2}$ be its second positive zero. Define $\beta(\alpha)<\alpha+1$ by $$ \int_0^{j_{\alpha,2}}u^{-\beta(\alpha)}J_\alpha(u)\,du=0. $$ We prove that $\beta''(\alpha)>0$ for $-1<\alpha\leq1/2$, including the left second deriv...
For a real transcendental number $\xi$, let $\omega_n^*(\xi)$ denote the supremum of all $\omega$ for which there exist infinitely many real algebraic numbers $\alpha$ of degree $\leq n$ satisfying $|\xi-\alpha|\leq H(\alpha)^{-\omega -1}$, where $H(\alpha)$ is the naive height of the minimal polynomial of $\alpha$. A...
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
Let $m\geq 1$ be a fixed integer, $a$ an integer satisfying $(a,m)=1$, and $z\geq 1$ a real parameter. Denote by $\omega_{z}(n;m,a)$ the number of distinct prime divisors $p$ of $n$ satisfying $p\equiv a\, (m)$ and $p\leq z$. We study an asymptotic behaviour of $\sum_{n\leq x}\left(\omega_{z}(n;m,a)-\frac{1}{\varphi(m)...
T. Minamide, Haruka Sakai, Y. Tanigawa· 0 citations
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...
Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves...
Zi-Kang Deng· 0 citations
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