We give new estimates for multiple exponential sums, which infers $$A(x)=C_1x+C_2x^{1/2}+C_3x^{1/3}+O(x^{1/4}e^{V(x)}), V(x)=\frac{1}{\sqrt{3}}(L\log{L})^{1/2}+O((L\log{L})^{1/2}),$$ where $L=\log{x}$, $A(x)$ is the number of non-isomorphic abelian groups of orders $\leq{x}$, and $x$ is large.

In this paper we establish some $L^p$ inequalities for polynomials having no zeros in $|z|<k$, where $k\ge 1$. Our results not only generalizes some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure.

We show that any $L$-function attached to a non-exceptionnal Hecke Grossencharakter $\Xi$ may be approximated by a truncated Euler product when $s$ lies near the line $\Re s=1$. This leads to some refined bounds on $L(s,\Xi)$.

By splitting bilateral series into two unilateral series, we derive several reciprocal formulae from Ramanujan's $_1\psi_1$ and Bailey's $_6\psi_6$ series identities, which generalize the reciprocity theorems due to Ramanujan and Andrews (1981).

Let $K$ be an algebraic field of degree $N$ and let $p$ be an odd prime. It is shown that if $K$ does not contain $p$-th primitive roots of unity and $f(X)=X^{p^k}+c$ with $k\ge1$ and non-zero $c\in K$, then the length of cycles of $f$ in $K$ is bounded by a value depending only on $K$ and $p$. If $p>2^N$, then this bound depends only on $N$.

We study growth envelopes of function spaces $B^s_{p,q}(\mathbb{R},w)$ and $F^s_{p,q}(\mathbb{R},w)$ where the weight belongs to some Muckenhoupt class $w\in\mathcal{A}_\infty$. This essentially extends partial forerunners in [13--16]. We also indicate some applications of these results.