Related papers: New proof of Weyl's theorem

We computationally verify that if $L(s,\chi)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10^{10}$ then $L(\sigma,\chi) \neq 0$ for real $\sigma \ge 1-1/(5\log q)$. The number of verified moduli exceeds benchmarks due to Watkins…

Let $\mu(n)$ denote the M\"obius function, define $M(x)= \sum_{n\leq x}^{}\mu (n)$. The main result of this paper is to prove that \begin{equation*} \displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*} which is equivalent to…

Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L…

In the early 1960s, P. Malliavin and L. A. Rubel gave a complete description of pairs of distributions of positive points $Z$ and $W$ such that for each entire function of exponential type $g\neq 0$ that vanishes on $W$, there is an entire…

In 1990, Daubechies proved a fundamental identity for Weyl-Heisenberg systems which is now called the Weyl-Heisenberg Frame Identity. WH-Frame Identity: If $g\in W(L^{\infty},L^{1})$, then for all continuous, compactly supported functions f…

Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire $L$-function of real archimedian type has a zero in the interval $\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\approx 14.13$ corresponds to the first…

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t))\\…

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral…

We give a simple proof of a fairly flexible comparison theorem for equations of the type $-(p(u'+su))'+rp(u'+su)+qu=0$ on a finite interval where $1/p$, $r$, $s$, and $q$ are real and integrable. Flexibility is provided by two functions…

We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl-Titchmarsh m-functions, $m_j(z)$, of two Schr\"odinger operators $H_j = -\f{d^2}{dx^2} + q_j$, j=1,2 in $L^2 ((0,R))$, $0<R\leq \infty$, are…

In this paper, using a basic algebraic approach, we demonstrate that, if the integers $x,y$ and $z$ are pairwise relatively prime, and $x^p-my^p=zw$ where $m$ is an integer and $p$ is a prime, then we can write $x,y,m,z$ and $w$ as explicit…

We study Liouville type of theorems for the Navier-Stokes and the Euler equations on $\Bbb R^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$ and $\int_{\Bbb…

We obtain a Li-Yau-type estimate for nonnegative ancient solutions to the subcritical semilinear heat equation $\frac{\p u}{\p t}=\De u+u^p$ in $\rz^n\times(-\infty,0)$. Then, we combine the Li-Yau type estimate and Melre-Zaag's result to…

Let $f $ be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L…

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important…

We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq 2\sqrt[3]{2},$…

We give a~new proof of the known criteria for the inequality \begin{equation*} \left(\int_{0}^{\infty}\left(\int_{0}^{t}f\right)^{q}w(t)\,dt\right)^{\frac{1}{q}} \leq C \left(\int_{0}^{\infty}f^{p}v\right)^{\frac{1}{p}}. \end{equation*} The…

We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies $ p\in L^1 (0,T; L^1 (\Bbb R^N))$ with $\int_{\Bbb R^N} p(x,t)dx \geq 0$, then the corresponding…

In 1996, H. Volkmer observed that the inequality \[(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx \] is satisfied with some positive constant $K>0$ for a certain class of…

A longstanding open problem in mathematical physics has been that of finding an action principle for the Einstein-Weyl (EW) equations. In this paper, we present for the first time such an action principle in three dimensions in which the…