Related papers: On the Asymptotics of a Prime Spin Relation

This article examines the value distribution of $S_{N}(f, \alpha) := \sum_{n=1}^N f(n\alpha)$ for almost every $\alpha$ where $N \in \mathbb{N}$ is ranging over a long interval and $f$ is a $1$-periodic function with discontinuities or…

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

Let $K$ be a number field with the discriminant $D_K$ and the class number $h_{K}$, which has bounded degree over $\mathbb{Q}$. By assuming GRH, we prove that every ideal class of $K$ contains a prime ideal with norm less than…

A sharp asymptotic formula for the sum of reciprocals of $\pi(n)$ is derived, where $\pi(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivi\'c and L. Panaitopol.

Let $\phi(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(\phi^n(0))_{n=0}^\infty$, which is the orbit of $0$ under iteration by $\phi$. Let $D_{d,c}$ denote the set of positive integers $n$ for…

We prove the spin-statistics theorem for massive particles obeying braid group statistics in three-dimensional Minkowski space. We start from first principles of local relativistic quantum theory. The only assumption is a gap in the mass…

We study a general class of weighted shifts whose weights $\alpha$ are given by $\alpha_n = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $N$ and $D$ are parameters so that $(N,D) \in (-1, 1)\times (-1, 1)$. Some few examples of these…

Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the…

A celebrated theorem of Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number $p$, we prove that there is an…

The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…

Let $K = \Q(\theta)$ be an algebraic number field with $\theta$ satisfying an irreducible polynomial $x^{9} - a$ over the field $\Q$ of rationals and $\Z_K$ denote the ring of algebraic integers of $K$. In this article, we provide the exact…

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta…

Radiative transitions gamma + Delta(1232) -> N^* are discussed in the nonrelativistic quark model with spin-orbit corrections for the 70-plet L^P=1^- nucleon resonances N^*. The reaction gamma + N -> gamma + Delta is considered as a tool to…

In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it, then $\sigma(N/q^{\alpha})/q^{\alpha}>5$. In general, we also show that if $\sigma(N/q^{\alpha})/q^{\alpha}<K$, where $K$…

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let $\ell$ be a rational prime and $K$ a rational function field $\Bbb F_q(t)$ with $\ell \nmid q$.…

There are two outstanding questions about division algebras of prime degree $p$. The first is whether they are cyclic, or equivalently crossed products. The second is whether the center, $Z(F,p)$, of the generic division algebra $UD(F,p)$…

We study the formula complexity of the word problem $\mathsf{Word}_{S_n,k} : \{0,1\}^{kn^2} \to \{0,1\}$: given $n$-by-$n$ permutation matrices $M_1,\dots,M_k$, compute the $(1,1)$-entry of the matrix product $M_1\cdots M_k$. An important…

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Poschl-Teller (tPT) potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number…

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence: $$\sum_{0<i_1<...<i_n<p}(i_1/3)(-1)^{i_1}/(i_1...i_n)=0 (mod p).$$