Related papers: Representation of even Gaussian integer \`a la Che…

Let $\Gamma$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $\Gamma$. Then…

Let $\mathcal{B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets of positive integers. Let $g_{\mathcal{B} }(n)$ count the number of representations of $n$ in the form $n = b_1\cdots b_h$, where $b_i \in B_i$ for all $i \in \{1,\ldots, h\}$.…

In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier…

The Gaussian integral, denoted as \( \int_{-\infty}^{\infty} e^{-x^2} dx \), plays a significant role in mathematical literature. In this paper, we explore a family of integrals related to Gaussian functions. Specifically, we introduce…

A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime…

In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.

Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…

We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of…

Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime.…

We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.

In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\le \ell_1 \le \ell_2$ be two integers, $\Lambda$ be the von Mangoldt function and…

Chung and Graham introduced a method to uniquely represent each positive integer using even-indexed Fibonacci terms. We generalize this result to represent each positive integer using other Fibonacci terms with equally-spaced indices.

We prove some special cases of Bergeron's inequality involving two Gaussian polynomials (or $q$-binomials).

In this paper, we use the transference principle to investigate the representation of sufficiently large positive integers as the sum of prime powers and integer powers, where the primes are drawn from a positive density subset of the set…

We study analogies between the rational integers on the real line and the Gaussian integers on other lines in the complex plane. This includes a Gaussian analog of Bertrands Postulate, the Chinese Remainder Theorem, and the periodicity of…

To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

For any given integer $k\geq 2$ we prove the existence of infinitely many $q$ and characters $ \chi\pmod q$ of order $k$, such that $|L(1,\chi)|\geq (e^{\gamma}+o(1))\log\log q$. We believe this bound to be best possible. When the order $k$…

In this article we show that the form $x^2 + iy^2 + z^2 + iw^2$ represents all gaussian integers. The main tools used in this proof are Fermat's little theorem (over finite field extensions), the Mordell-Niven theorem (representation of…

In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…