Papers by Es-said En-naoui
This article extends our previous study [5] on the summatory behavior of Euler's totient function... more This article extends our previous study [5] on the summatory behavior of Euler's totient function φ(n). We investigate two complementary restricted sums, Υ(x, p) = k≤x gcd(k,p)=1 φ(k), ∆(x, p) = k≤x p|k φ(k), which satisfy the decomposition Ψ(x) = Υ(x, p) + ∆(x, p), where Ψ(x) = k≤x φ(k). We establish recurrence formulas, congruence relations, and generating function identities for ∆(x, p). In particular, we prove that ∆(x, p) ≡ 0 (mod p-1) for every prime p, and derive the asymptotic expansion ∆(x, p) = 3 π 2 (p + 1) x 2 + O(x log x). Furthermore, we study average orders, connections with ω(n), and relations with divisor structures. These results refine the analytic understanding of totients in arithmetic progressions and complement the asymptotic theory of Ψ(x).
Proof Gooldbach , 2025
This article introduces Symmetric Primality Axiomatics (SPA), a self-contained firstorder axiomat... more This article introduces Symmetric Primality Axiomatics (SPA), a self-contained firstorder axiomatic theory extending basic arithmetic by a unary predicate P(x) (interpreted as "x is prime (in SPA)"). SPA contains a small list of explicit axioms about P (not derived from classical prime theory) designed so that the main centered-prime statement ∀n > 4 ∃k (P(n-k) ∧ P(n + k)) is provable within SPA. The article gives precise formal axioms, proves all intermediary lemmas inside SPA, and presents a complete derivation of the principal theorem from those axioms. The exposition is entirely internal to SPA and avoids appeal to any external theorems about classical primes.
Euler's totient function counts the positive integers up to a given integer n that are relatively... more Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum : n k=1 p|k φ(k) , for every prime number p .
arXiv (Cornell University), Dec 11, 2023
The Additive Transform of an arithmetic function represents a novel approach to examining the int... more The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate new arithmetic functions by combining the values of an existing function under an additive operation. The resulting framework not only extends our understanding of classical arithmetic functions but also provides a versatile tool for exploring additive relationships within the realm of number theory. In this article, we present the fundamental principles of the Additive Transform and illustrate its application through various examples, shedding light on its potential implications for diverse mathematical domains. For all positive integer n. a motivation for the present study is to give a new concept named the Additive transform of an arithmetic function f when f equals some special arithmetic functions, that new concept can help us to prove many results like : µ * f.Id (n) = ϕ(n)f (n) + ϕ(n) p α ||n f (p α) − f (p α−1) p − 1 (1) where f is an additive function .
Additive transform of an arithmetic function : Part I, 2023
The Additive Transform of an arithmetic function represents a novel approach to examining the int... more The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate new arithmetic functions by combining the values of an existing function under an additive operation. The resulting framework not only extends our understanding of classical arithmetic functions but also provides a versatile tool for exploring additive relationships within the realm of number theory. In this article, we present the fundamental principles of the Additive Transform and illustrate its application through various examples, shedding light on its potential implications for diverse mathematical domains. For all positive integer n. a motivation for the present study is to give a new concept named the Additive transform of an arithmetic function f when f equals some special arithmetic functions, that new concept can help us to prove many results like : µ * f.Id (n) = ϕ(n)f (n) + ϕ(n) p α ||n f (p α) − f (p α−1) p − 1 (1) where f is an additive function .
arXiv (Cornell University), Dec 11, 2022
The main object of this paper is to give the generalized von mangoldt function using the L-additi... more The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative δ and Dirichlet series defined by: n≥1 f (n)δ(n) n s where f is a classical arithmetic function.
Study of the generalized von mangoldt function defined by L-additive function
arXiv (Cornell University), Jan 23, 2023
The main object of this paper is to study the generalized von mangoldt function using the L-addit... more The main object of this paper is to study the generalized von mangoldt function using the L-additive function, which can help us give many result about the classical arithmetic function.
arXiv: General Mathematics, 2019
We define the derivative of an integer to be the map sending every prime to 1 and satisfying the ... more We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibniz rule. The aim of this article is to calculate the Dirichlet product of this map with a function arithmetic multiplicative.
The Goldbach conjecture dates back to 1742 ; we refer the reader to [1]-[2] for a history of the ... more The Goldbach conjecture dates back to 1742 ; we refer the reader to [1]-[2] for a history of the conjecture. Christian Goldbach stated that every odd integer greater than seven can be written as the sum of at most three prime numbers. Leonhard Euler then made a stronger conjecture that every even integer greater than four can be written as the sum of two primes. Since then, no one has been able to prove the Strong Goldbach Conjecture. The only best known result so far is that of Chen [3], proving that every suciently large even integer N can be written as the sum of a prime number and the product of at most two prime numbers. Additionally, the conjecture has been veried to be true for all even integers up to 4.10 18 in 2014 , Jërg [4] and Tomás [5]. In this paper, we prove that the conjecture is true for all even integers greater than 8.
Euler’s totient function counts the positive integers up to a given integer n that are relatively... more Euler’s totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum : n ∑ k=1 p|k φ(k) , for every prime number p .
Es-said En-naoui, 2023
The main object of this paper is to study the generalized von mangoldt function using the L-addit... more The main object of this paper is to study the generalized von mangoldt function using the L-additive function, which can help us give many result about the classical arithmetic function.
Es-said En-naoui, 2023
The main object of this paper is to give the generalized von mangoldt function using the L-additi... more The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative δ and Dirichlet series defined by: n≥1 f (n)δ(n) n s where f is a classical arithmetic function.
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Drafts by Es-said En-naoui
Es-said En-naoui, 2023
In this paper we give formula of the Dirichlet products of two arithmetic functions f and g with ... more In this paper we give formula of the Dirichlet products of two arithmetic functions f and g with f and g is multiplicative or additive , and we give some of our results about the function f and g where f and g is one of the following classical arithmetic functions: Euler's totient function ϕ,the number of distinct prime factors ω,the prime factor counting functions Ω, the identity function , the sum-of-divisors function σ, the divisor function τ , the Mobius function µ ,... The aim of this article is to give the result of the Dirichlet Series of the En-naoui function defined by : Φ ϕ (n) = n p|n 1 − 1 p , and also prove again my result on my first article about arithmetic derivative function (see [1])
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Papers by Es-said En-naoui
Drafts by Es-said En-naoui