Weakly compactly generated Banach spaces and their subspaces are characterized by the presence of projectional skeletons with some additional properties. We work with real spaces. However the presented statements can be extended, without much extra effort, to complex spaces.

[EN] While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of infinite-dimensional separable quotients in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the presented results are consequences of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c_0 and lp. Also recent results of Argyros, Dodos, Kanellopoulos [see 1 in the paper] and Sliwa [see 64 in the paper] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica.

Let $\phi$ denote the Euler totient function, defined by $id*\mu$ where $\mu$ is the M\{o}bius function. We shall consider the $k$-th Riesz mean of the arithmetical function $n/\phi(n)$ for any positive integer $k \geq 2$ on the assumption of the Riemann Hypothesis. Our result is a refinement of Theorem 2 in A. Sankaranarayanan and S.K. Singh [6]. We also improve it upon the assumption of the Gonek-Hejhal Hypothesis.

Taking the Iwaniec explicit formula as a starting point, we bound the exponent in the error term of the prime geodesic theorem for the modular surface to $2/3$, outside a set of finite logarithmic measure.

Let $F(z)$ be a Hecke-Maass form for $SL(m,\mathbb{Z})$ and $A_F(n,1, \dots, 1)$ be the coefficients of $L$-function attached to $F.$ We study the cancellation of $A_F(n,1, \dots, 1)$ for twisted with a nonlinear exponential function at primes, namely the sum \begin{equation*} \sum_{n \leq N} \Lambda (n)A_F(n,1, \dots, 1)e ( \alpha n^\theta ), \end{equation*} where $0<\theta<2/m$. We also strengthen the corresponding previous results for holomorphic cusp forms for $SL(2,\mathbb{Z}),$ and improve the estimates of Ren-Ye on the resonance of exponential sums involving Fourier coefficients of a Maass form for $SL(m,\mathbb{Z})$.

Inspired by the work of Silverman on the geometry and the arithmetic of monomial maps and also on the translated maps on Abelian varieties, we generalize his results to the case of the translated monomial maps.

In this paper, we study the diophantine equation $$y^{p} = f(x_{1}, x_{2}, ..., x_{r}),$$ where $f(x_{1}, x_{2}, ..., x_{r})$ is a real polynomial in variables $x_{1}, x_{2}, ..., x_{r}$ in $R$, a group of real numbers under the usual addition $+$, having the least element property.

Let $L$ be a finite Galois extension of the number field $K$. We unconditionally bound the least prime ideal of $K$ occurring in the Chebotarev Density Theorem as a power of the discriminant of $L$ with an explicit exponent. We also establish a quantitative Deuring-Heilbronn phenomenon for the Dedekind zeta function.

In [Ru15], the author introduced the notion of Nevanlinna constant (denoted by $Nev(D)$) for any effective Cartier divisor $D$ on a normal projective variety $X$, and established a defect relation for Zariski-dense holomorphic mappings $f: {\mathbb C}\rightarrow X$ in terms of $Nev(D)$. In this paper, we prove its counterpart result in Diophantine approximation, according to Vojta's correspondence (or Vojta's dictionary [Voj87]). The results obtained gave the quantitative extension of the earlier results of Corvaja-Zannier [CZ04a,CZ04b], Evertse-Ferretti [EF02, EF08], A. Levin [Lev09], P. Autissier [Aut1], and others.

The paper gives sufficient conditions to have the equality between two diametral dimensions of metrizable locally convex spaces and examples of Kothe echelon spaces satisfying them. It also provides examples for which the equality does not hold.