Let $2\leq p<\infty $ and X be a complex infinite-dimensional Banach space. It is proved that if X is p-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space $\mathscr {H}^{\text {weak}}_p(X)$ to the Hardy space $\mathscr {H}^+_p(X)$ of vector-valued Dirichlet series. To obtain this, a Littlewood–Paley inequality for Dirichlet series is established.

We give a new proof of a theorem of Bell and Coons [‘Transcendence tests for Mahler functions’, Proc. Amer. Math. Soc. 145(3) (2017), 1061–1070] on the leading order radial asymptotics of Mahler functions that are the generating functions of regular sequences. Our method allows us to provide a description of the oscillations whose existence was shown by Bell and Coons. This extends very recent results of Poulet and Rivoal [‘Radial behavior of Mahler functions’, Int. J. Number Theory, to appear].

In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$. In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$, thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$, where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two.

In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach space X. In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove that every equivalence class is sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class, i.e., the condition of almost periodicity of a function $f\in AP(\mathbb {R},X)$ yields that every sequence of translates of f has a subsequence that converges to a function equivalent to f. This extends previous work by the same authors on the case of numerical almost periodic functions.

We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class.

Müntz-Legendre polynomials ${{L}_{n}}\left( \Lambda ;\,x \right)$ associated with a sequence $\Lambda \,=\,\left\{ {{\lambda }_{k}} \right\}$ are obtained by orthogonalizing the system $\left( {{x}^{{{\lambda }_{0}}}},{{x}^{{{\lambda }_{1}}}},{{x}^{{{\lambda }_{2}}}},... \right)$ in ${{L}_{2}}\left[ 0,1 \right]$ with respect to the Legendre weight. If the ${{\lambda }_{k}}\text{ }\!\!'\!\!\text{ s}$ are distinct, it is well known that ${{L}_{n}}\left( \Lambda ;\,x \right)$ has exactly $n$ zeros ${{l}_{n,n}}\,<\,{{l}_{n-1,n}}\,<\,\cdot \cdot \cdot \,<\,{{l}_{2,n}}\,<\,{{l}_{1,n}}$ on $\left( 0,1 \right)$.

First we prove the following global bound for the smallest zero,

$$\exp \left( -4\sum\limits_{j=0}^{n}{\frac{1}{2\text{ }\!\!\lambda\!\!\text{ j}\,\text{+}\,\text{1}}} \right)\,<\,{{l}_{n,n}}.$$

An important consequence is that if the associated Müntz space is non-dense in ${{L}_{2}}\left[ 0,1 \right]$, then

$$\underset{n}{\mathop{\inf }}\,\,{{x}_{n,n}}\,\ge \,\exp \,\left( -4\,\sum\limits_{j=0}^{\infty }{\frac{1}{2{{\text{ }\!\!\lambda\!\!\text{ }}_{j}}\,+\,1}} \right)\,>\,0,$$

so the elements ${{L}_{n}}\left( \Lambda ;\,x \right)$ have no zeros close to 0.

Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed,

$$\underset{n\to \infty }{\mathop{\lim }}\,\,\left| \log \,{{l}_{k,n}} \right|\,\sum\limits_{j=0}^{n}{\left( 2{{\text{ }\!\!\lambda\!\!\text{ }}_{j}}\,+\,1 \right)}\,=\,{{\left( \frac{jk}{2} \right)}^{2}},$$

where ${{j}_{k}}$ denotes the $k$-th zero of the Bessel function ${{J}_{0}}.$

We provide concise series representations for various $\text{L}$-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying $\text{L}$-series.