Let $f\,=\,\sum{_{n=1}^{\infty }{{a}_{f}}\left( n \right){{q}^{n}}}$ be a cusp form with integer weight $k\,\ge \,2$ that is not a linear combination of forms with complex multiplication. For $n\,\ge \,1$, let

$${{i}_{f}}\left( n \right)\,=\,\left\{ _{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise}\text{.}}^{\max \left\{ i\,:\,{{a}_{f}}\left( n+j \right)\,=\,0\,\text{for}\,\text{all}\,0\,\,\le \,j\,\le \,i \right\}\,\,\,\,\,\,\,\text{if}\,{{\text{a}}_{f}}\left( n \right)\,=\,0,} \right.\,$$

Concerning bounded values of ${{i}_{f}}\left( n \right)$ we prove that for $\in \,>\,0$ there exists $M\,=\,M\left( \in ,\,f \right)$ such that $\#\left\{ n\,\le \,x\,:\,{{i}_{f}}\left( n \right)\,\le \,M \right\}\,\ge \,\left( 1-\in \right)x.$ Using results of Wu, we show that if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then ${{i}_{f}}\left( n \right)\,{{\ll }_{f,\in }}\,{{n}^{\frac{51}{134}+\in }}$. Using a result of David and Pappalardi, we improve the exponent to $\frac{1}{3}$ for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate ${{i}_{f}}\left( n \right)$ on the average using well known bounds on the Riemann Zeta function.

In the paper we formulate a Covering Property Axiom, $\text{CP}{{\text{A}}_{\text{prism}}}$, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.

(a) There exists a family $\mathcal{F}$ of less than continuum many ${{C}^{1}}$ functions from $\mathbb{R}$ to $\mathbb{R}$ such that ${{\mathbb{R}}^{2}}$ is covered by functions from $\mathcal{F}$, in the sense that for every $\left\langle x,\,y \right\rangle \,\in \,{{\mathbb{R}}^{2}}$${{\mathbb{R}}^{2}}$ there exists an $f\,\in \,\mathcal{F}$ such that either $f\left( x \right)\,=\,y$ or $f\left( y \right)\,=\,x$.

(b) For every Borel function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many “${{C}^{1}}$” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$.

(c) For every $n\,>\,0$ and a ${{D}^{n}}$ function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many ${{C}^{n}}$ functions whose graphs cover the graph of $f$.

We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiî.

With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F. V. Atkinson.

Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, ${{e}^{-tA}}$, can be bounded below from ${{L}^{p}}$ to ${{L}^{q}}$ when $p,\,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.

In this paper we make explicit all $L$-functions in the Langlands–Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local $L$-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for $\operatorname{Re}\left( s \right)\,\ge \,1/2$ in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrum and determining poles of automorphic $L$-functions.

Differential operators ${{D}_{x,}}\,{{D}_{y}}$, and ${{D}_{z}}$ are formed using the action of the 3-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L\,=\,D_{x}^{2}+D_{y}^{2}+D_{z}^{2}$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space ${{L}^{2}}\left( S \right)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

In this paper we describe reducibility of non-unitary generalized principal series for classical $p$-adic groups in terms of the classification of discrete series due to Mœglin and Tadić.

We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL\left( 2,\,k \right)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL\left( 2,\,k \right)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.