Some Pythagorean type equations concerning arithmetic functions

Let be a positive integer and be the sum of the digits in basis q of the positive integer n. We prove that the quotient has a normal order one, where and are respectively, the number of distinct prime factors and the number of prime factors p of a positive integer n counted with multiplicity such that mod Moreover, we discuss sums of the form where f is a multiplicative function.

Let [Formula: see text] be a positive integer, and define [Formula: see text] for [Formula: see text], where [Formula: see text] denotes the number of distinct prime factors of [Formula: see text], and [Formula: see text] represents the total number of prime factors of [Formula: see text] (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that [Formula: see text] which is similar to the known identity [Formula: see text] equivalent to the Prime Number Theorem. For [Formula: see text], we prove that [Formula: see text] We also raise a hypothesis on the parities of [Formula: see text] which implies the Riemann Hypothesis.

The set of the positive integers and positive square—free integers are denoted by IN and IN*, respectively, and we write IN(n) = IN ∩ [1, n], IN* (n) = IN* ∩ [1, n], where [1, n] = {1, 2, ... , n}. The set of primes is denoted by P. The smallest and greatest prime factors of the positive integer n are denoted by p(n) and P(n), respectively. ω(n) denotes the number of distinct prime factors of n, while Ω(n) denotes the number of prime factors of n counted with multiplicity: $$\omega \left( n \right)\, = \,\sum\limits_{p|n} {1,} \,\Omega \left( n \right)\, = \,\sum\limits_{{p^\alpha }\parallel n} {\alpha .} $$ µ(n) denotes the Mobius function.

For a given number of resistance alms n, and capacitance arms nc in a symmetrical parallel RC multiple-T, it is possible to know the maximum number of dissimilar resonant frequency combinations theoretically from a formula derived using a combinational technique. Verification of the results is done by counting the number of dissimilar frequencies for the ratio nr/ nc. WM=WD (nrnc)1/2 and also from frequencies practically determined. The technique common to both these methods is the combinatorial nne as used in deriving Euler's Totient function in the theory of numbers.

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

Multicast is an important communication method used in many emerging multimedia networking applications. Providing efficient security method to support the distribution of multimedia multicast is a challenging issue, since the group membership in such applications requires dynamic key generation and updation which takes more computation time. Moreover, the key must be sent securely to the group members. In this paper, we propose a new Key Distribution Protocol that provides more security and also reduces computation complexity. To achieve higher level of security, we use Euler's Totient Function φ(n) in the key distribution protocol. Therefore, it increases the key space while breaking the re-keying information. Two major operations in this scheme are join and leave operations for managing group memberships. An N-ary tree is used to reduce number of multiplications needed to perform the member leave operation. Using this tree, we reduce the computation time when compared with the existing key management schemes.

This letter introduces a real valued summation known as Complex Conjugate Pair Sum (CCPS). The space spanned by CCPS and its one circular downshift is called {\em Complex Conjugate Subspace (CCS)}. For a given positive integer $N\geq3$, there exists $\frac{\varphi(N)}{2}$ CCPSs forming $\frac{\varphi(N)}{2}$ CCSs, where $\varphi(N)$ is the Euler's totient function. We prove that these CCSs are mutually orthogonal and their direct sum form a $\varphi(N)$ dimensional subspace $s_N$ of $\mathbb{C}^N$. We propose that any signal of finite length $N$ is represented as a linear combination of elements from a special basis of $s_d$, for each divisor $d$ of $N$. This defines a new transform named as Complex Conjugate Periodic Transform (CCPT). Later, we compared CCPT with DFT (Discrete Fourier Transform) and RPT (Ramanujan Periodic Transform). It is shown that, using CCPT we can estimate the period, hidden periods and frequency information of a signal. Whereas, RPT does not provide the frequency information. For a complex valued input signal, CCPT offers computational benefit over DFT. A CCPT dictionary based method is proposed to extract non-divisor period information.

On greatest common divisor matrices and their applications

Round numbers revisited. A new fuzzy approach

Let \({q\geqq2}\) be an integer and denote S(n) the sum of the digits in base q of the positive integer n. Our main result is to estimate the sum \({\Sigma_{n\leqq x}\tilde{\omega}(n)}\) where \({\tilde{\omega}(n)}\) is the number of distinct prime factors p of n such that \({S(p) \equiv a \,{\rm mod} \,b \,(a \in \mathbb{Z}, b\geqq 2)}\).

On the normal number of prime factors of φ(n) subject to certain congruence conditions

In this paper, we establish a quite general mean value result of arithmetic functions over short intervals with the Selberg-Delange method and give some applications. In particular, we generalize Selberg's result on the distribution of integers with a given number of prime factors and Deshouillers-Dress-Tenenbaum's arcsin law on divisors to the short interval case.

<p>In this paper, we introduce a new arithmetic function r sp(n) which we called the<br /> simple divisor function. The main purpose of this paper is to study the asymptotic properties of the mean value of r sp(n) by using the elementary methods, and obtain an interesting asymptotic formula for it.</p>

On the distribution of integers with a given number of prime factors