Related papers: Discrepancies in the distribution of Gaussian prim…

Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with…

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density…

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

In order to study signed Eulerian numbers, we introduce permutations of a particular type, called parity-alternate permutations, because they take even and odd entries alternately. The objective of this paper is twofold. The first is to…

The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…

For a positive integer $n$, if $\sigma(n)$ denotes the sum of the positive divisors of $n$, then $n$ is called a deficient perfect number if $\sigma(n)=2n-d$ for some positive divisor $d$ of $n$. In this paper, we prove some results about…

Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to…

In this paper, it is established that every sufficiently large positive integer $n$ subject to $n\equiv0\pmod2$ can be represented as a sum of one square of prime and seventeen fifth powers of primes, which gives an enhancement upon the…

"Goldbach's Conjecture" proven by analysis of how all combinations of the odd primes, summed in pairs, generates all of the even numbers.

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a…

We study differentiability properties of functions defined in the euclidean space in terms of a conical square function which is analogue to the classical square function introduced by Stein and Zygmund in the sixties. Pointwise…

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus $q$, any two reduced congruence classes $a_1$ and $a_2$ mod $q$, and any $r_1,r_2 \ge 1$, a positive…

Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio…

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive…

A representation number is a function which expresses the number of ways an integer can be written as a sum of elements of chosen sets. One of the oldest number-theoretic results on representation numbers is Fermat's theorem which says that…

Let $\lambda(n)$ be the Liouville function. We study the distribution of \[ \frac{1}{x^{1/2}}\sum_{x\leq n\leq 2x}\lambda(f(n)) \] over random polynomials $f$ of fixed degree $d$ and coefficients bounded in magnitude by $H$. In particular…

We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…

Erd\H{o}s and Hall studied the angular distribution of Gaussian integers with a fixed norm. We generalize their result to the angular distribution of integral ideal numbers with a fixed norm in any quadratic extension.

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$.…

We study the distribution of primes from a topological viewpoint. Certain conjecture is introduced, and we show that it is equivalent to the Riemann Hypothesis.