Upper bound estimates for the exponential sum $$\sum_{K<\kappa_j\le K^\prime<2K} \alpha_j H_j^3(\tfrac{1}{2}) \cos(\kappa_j\log(\frac{4eT}{\kappa_j}))\qquad(T^\varepsilon \le K \le T^{1/2-\varepsilon})$$ are considered, where $\alpha_j = |\rho_j(1)|^2(\cosh\pi\kappa_j)^{-1}$, and $\rho_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $\lambda_j = \kappa_j^2 + \tfrac{1}{4}$ to which the Hecke series $H_j(s)$ is attached. The problem is transformed to the estimation of a classical exponential sum involving the binary additive divisor problem. The analogous exponential sums with $H_j(\tfrac{1}{2})$ or $H_j^2(\tfrac{1}{2})$ replacing ${H_j^3(\tfrac{1}{2})}$ are also considered. The above sum is conjectured to be $\ll_\varepsilon K^{3/2+\varepsilon}$, which is proved to be true in the mean square sense.

In this note we investigate problems related to the unique factorization of some semigroups of classical $L$-functions. The semigroups of Artin and automorphic $L$-functions as well as the semigroup generated by the Hecke $L$-functions of finite order are studied. The main result of the paper shows that in the latter semigroup the unique factorization into primitive elements does not hold. This closes a possible way of attacking the famous Dedekind conjecture concerning the divisibility of the Dedekind zeta functions.

Establishing the structure of dense sum-free subsets of the torus group $\mathbb{R}/\mathbb{Z}$, we find an absolute constant $\alpha_0<1/3$ such that for any sum-free subset $A\subseteq\mathbb{R}/\mathbb{Z}$ with the inner measure $\mu(A)>\alpha_0$ there exists an integer $q\ge 1$ so that $$A \subseteq \bigcup_{j=0}^{q-1}[ \frac{j+\mu(A)}q, \frac{j+1-\mu(A)}q ].$$

Let $\mathbb{J}_q^k[t]$ denote the additive closure of the set of $k$th powers in the polynomial ring $\mathbb{F}_q[t]$, defined over the finite field $\mathbb{F}_q$ having $q$ elements. We show that when $s\ge k+1$ and $q \ge k^{2k+2}$, then every polynomial in $\mathbb{J}_q^k[t]$ is the sum of at most $s$ $k$th powers of polynomials from $\mathbb{F}_q[t]$. When $k$ is large and $s \ge (\frac{4}{3}+o(1)) k\log k$, the same conclusion holds without restriction on $q$. Refinements are offered that depend on the characteristic of $\mathbb{F}_q$.

We address a problem initiated by Davenport in 1937 ([5] et [6]). Let $z\in\mathbb{C}$. We study the conditions on real $\vartheta$, and its continued fraction expansion, for the validity of the formal identity $$\sum_{m=1}^\infty{\tau_{z+1}(m) \over \pi m}\sin(2 \pi m\vartheta)+\sum_{n=1}^\infty{\tau_z(n) \over \pi n}B(n\vartheta)=0$$ where $B$ denotes the first Bernoulli function and $\tau_z$, the Piltz function of order $z$. We use methods developed by Fouvry, La Bret\`eche and Tenenbaum ([8] et [2]), based on summation over friable integers.

This article is a discussion of Cantor's famous diagonal proof showing the existence of transcendental numbers. We study three questions raised in a previous joint paper [1].

It is shown that a small change in the argument of Harper and Murty implies that there are at most two real quadratic fields with class-number one and without Euclidean algorithm.

Let $F(x_1,\ldots,x_m) = u_1 x_1 + \cdots + u_mx_m$ be a linear form with nonzero, relatively prime integer coefficients $u_1, \ldots, u_m$. For any set $A$ of integers, let $F(A)=\{F(a_1,\ldots,a_m): a_i \in A for i=1,\ldots,m\}.$ The {\it representation function} associated with the form $F$ is $$ R_{A,F}(n) = \card ( \{ (a_1,\ldots,a_m)\in A^m: F(a_1,\ldots, a_m) = n \} ). $$ The set $A$ is a {\it basis with respect to $F$ for almost all integers} if the set ${\bf Z} \setminus F(A)$ has asymptotic density zero. Equivalently, the representation function of a basis for almost all integers is a function $f:{\bf Z} \rightarrow {\bf N_0}\cup\{\infty\}$ such that $f^{-1}(0)$ has density zero. Given such a function, the inverse problem for bases is to construct a set $A$ whose representation function is $f$. In this paper the inverse problem is solved for binary linear forms. for binary linear forms.

It is proved that if $\mathcal{A}$ is a set of positive integers with $1\in\mathcal{A}$ then almost all partitions of $n$ into the elements of $\mathcal{A}$ contain the summand 1.

In the 1930's Cramér created a probabilistic model for primes. He applied his model to express a very deep conjecture about large differences between consecutive primes. The general belief was for a period of 50 years that the model reflects the true behaviour of primes when applied to proper problems. It was a great surprise therefore when Helmut Maier discovered in 1985 that the model gives wrong predictions for the distribution of primes in short intervals. In the paper we analyse this phenomen, and describe a simpler proof of Maier's theorem which uses only tools available at the mid thirties. We present further a completely different contradiction between the model and the reality. Additionally, we show that, unlike to the contradiction discovered by Maier, this new contradiction would be present in essentially all Cramér type models using independent random variables.

In this third part of our work, we go back to the study of the $\nu^{(k)}_G$ functions (introduced in the first one), which count the minimal cardinality of a sumset containing an element with a single representation. An upper bound for these functions is obtained in the case $k=2$ using what we call the generalized increasingly small sumsets property, which is proved to hold for all Abelian groups. Moreover, we show that our bound cannot be improved in general.

We provide some evidence that the eigenvalues of the hermitian form $\sum_{a/q}|\sum_{n\le N}\varphi_ne(na/q)|^2$ tend to have a limit distribution when $N$ and $Q$ go simultaneously to infinity in such a way that $N/Q^2$ tends to a constant. We also present some background material, as well as a large sieve equality, when $N\Log^7 N = o(Q)$, that follows from our results.

We define $n$-dimensional Beukers-type integrals over the unit hypercube. Using an $n$-dimensional birational transformation we show that such integrals are equal to suitable $n$-dimensional Sorokin-type integrals with linear constraints, and represent linear forms in $1, \zeta(2), \zeta(3), \dots, \zeta(n)$ with rational coefficients.

The length $L(P)$ of a polynomial $P$ is the sum of the absolute values of the coefficients. For $P\in\mathbb{R}[x]$ the properties of $l(P)$ are studied, where $l(P)$ is the infimum of $L(PG)$ for $G$ running through monic polynomials over $\mathbb{R}$.

For any unitary cuspidal representations $\pi_n$ of $GL_n(\mathbb{Q}_\mathbb{A})$, $n=2,3,4$, respectively, consider two automorphic representations $\Pi$ and $\Pi'$ of $GL_6(\mathbb{Q}_\mathbb{A})$, where $\Pi_p\cong\wedge^2\pi_{4,p}$ for $p\neq 2,3$ and $\pi_{4,p}$ not supercuspidal ($\pi_{4, p}$ denotes the local component of $\pi_4$), and $\Pi'=\pi_2\boxtimes\pi_3$. First, Hypothesis H for $\Pi$ and $\Pi'$ is proved. Then contributions from prime powers are removed from the prime number theorem for cuspidal representations $\pi$ and $\pi'$ of $GL_m(\mathbb{Q}_\mathbb{A})$ and $GL_{m'}(\mathbb{Q}_\mathbb{A})$, respectively. The resulting prime number theorem is unconditional when $m,m'\leq 4$ and is under Hypothesis H otherwise.