Related papers: A Totient Function Inequality
We present a simple, closed formula which gives all the primes in order. It is a simple product of integer floor and ceiling functions.
In the work the fair division problem for two participants in presence of both divisible and indivisible items is considered. The set of all divisions is formally described; it is demonstrated that fair (in terms of Brams and Taylor)…
In this paper, the authors establish some inequalities involving the $q$-extension of the classical Gamma function. These inequalities provide bounds for certain ratios of the $q$-extended Gamma function. The procedure makes use of…
The undecidability of the additive theory of primes (with identity) as well as the theory Th(N,+, n -> p\_n), where p\_n denotes the (n+1)-th prime, are open questions. As a possible approach, we extend the latter theory by adding some…
An inequality is derived for the correlation of two univariate functions operating on symmetric bivariate normal random variables. The inequality is a simple consequence of the Cauchy-Schwarz inequality.
In this paper our aim is to prove some monotonicity and convexity results for the modified Struve function of the second kind by using its integral representation. Moreover, as consequences of these results, we present some functional…
In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete…
This paper aims to characterize the function appearing in the weighted Hermite-Hadamard inequality. We provide improved inequalities for the weighted means as applications of the obtained results. Modifications of the weighted…
We first prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial…
In this paper we investigate a certain category of cotangent sums and more specifically the sum $$\sum_{m=1}^{b-1}\cot\left(\frac{\pi m}{b}\right)\sin^{3}\left(2\pi m\frac{a}{b}\right)\:$$ and associate the distribution of its values to a…
We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression.…
We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…
We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.
A composite number $n$ is called a Lehmer number when $\phi(n) | n - 1$, where $\phi$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $\phi(n)| n-1$? No such numbers are known.…
Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker…
The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in…
Euler's totient function, $\varphi(n)$, which counts how many of $0,1,\dots,n-1$ are coprime to $n$, has an explicit asymptotic lower bound of $n/\log \log n$, modulo some constant. In this note, we generalise $\varphi$; given an…
In this paper, new upper and lower bounds for the Trapezoid inequality of absolutely continuous functions are obtained. Applications to some special means are provided as well.
This article presents an equivalent formulation of the implicit complementarity problem. We demonstrate that solution of the equivalent formulation is equivalent to the solution of the implicit complementarity problem. Moreover, we provide…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein together with theorems corollaries, formulae, examples, mathematical criteria, etc. (about integer sequences, numbers, quotients, residues,…