We determine explicitly the structure of the torsion group over the maximal abelian extension of $\mathbb{Q}$ and over the maximal $p$-cyclotomic extensions of $\mathbb{Q}$ for the family of rational elliptic curves given by $Y^2 = X^3 + AX$, where $A$ is an integer.

In this paper we consider Diophantine Approximation using numbers which are a sum of two odd primes (Goldbach Numbers). We use exponential sums and sieve methods to establish, for any irrational $\alpha$ and arbitrary real $\beta$, that there are infinitely many solutions to \[ ||\alpha n + \beta|| < n^{- \frac56}, n = p_1+p_2, p_j \text{denotes an odd prime}, \] where $||\cdot||$ denotes, as usual, distance to a nearest integer.

By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $\geq 4$, and show the existence of infinitely many elliptic curves of rank $\geq 5$, parameterized by the points of an elliptic curve of positive rank.

We shall study some special infinite products obtained by modifying the product expression of the generating function of the Thue-Morse sequence. We first prove that these products are transcendental functions. Then the values of these functions at algebraic points are proved to be transcendental by applying Mahler's method to suitably constructed auxiliary functions. Finally, also irrationality exponents of the values at rational points are considered.

For an integer $m \geq 3$, does there exist an absolute constant $K(m)$ such that every polynomial with $m$ non-zero coefficients has an irreducible factor with at most $K(m)$ coefficients? A previous result in the literature establishes $K(3) \geq 9$, which is here improved to $K(3) \geq 12$. Improvements on known bounds are also given for $m=4,5,6$, and for $K(m)$, when $m \geq 7$.

Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields.

We continue our investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a ``short'' interval, whose admissible length depends on whether or not we assume the Riemann Hypothesis.

We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan [3] on quadratic Dirichlet character sums.

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.