Let $q\ge 3$ be an integer, let $\chi$ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac{\chi \left(n\right)}{{(n+a)}^s},$$ where $s=\sigma +\mathrm{i}t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac{1}{2}+\mathrm{i}t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.

Let Ω be a bounded domain in ${ℝ}^n$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^2\left(\Omega \right)$ into itself with compact inverse, with eigenvalues $-{\lambda }_i$, each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $Lu+b{u}^{+}-a{u}^{-}=f\left(x\right)$ in Ω, u=0 on ∂ Ω. We assume that $a<{\lambda }_1$, ${\lambda }_2<b<{\lambda }_3$ and f is generated by ${\varphi }_1$ and ${\varphi }_2$. We show a relation between the multiplicity of solutions and source terms...

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an $O\left(ϵ\right)$ estimate in $H^1$ for solutions with Dirichlet condition.

For $x\in ℂ$, $\left|x\right|\&lt;1$, $s\in \mathbb{N}$, let ${\mathrm{Li}}_s\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $\left|\alpha \right|\&lt;1$, the $\mathbb{Q}\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_1\left(\alpha \right),{\mathrm{Li}}_2\left(\alpha \right),\cdots$ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha$. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

Given an irreducible algebraic curves $A$ in ${ℂ}^N$, let $m_d$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$. Let $K$ be a nonpolar compact subset of $A$, and for each $d=1,2,...,$ choose $m_d$ points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques...

To a plurisubharmonic function $u$ on ${\mathbf{C}}^n$ with logarithmic growth at infinity, we may associate the Robin function $${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$ defined on ${\mathbf{P}}^{n-1}$, the hyperplane at infinity. We study the classes $L_{+}$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=log(1+|z\left|\right)+O\left(1\right)$ and (respectively) for which the function ${\rho }_u$ is not identically $-\infty$. We obtain an integral formula which connects the Monge-Ampère measure on the space ${\mathbf{C}}^n$ with the Robin function on ${\mathbf{P}}^{n-1}$. As an application we obtain a criterion...