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DIRICHLET PRODUCT of DERIVATIVE ARITHMETIC with an ARITHMETIC FUNCTION MULTIPLICATIVE Es-Said En-Naoui

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DIRICHLET PRODUCT of DERIVATIVE ARITHMETIC with an ARITHMETIC FUNCTION MULTIPLICATIVE Es-Said En-Naoui DIRICHLET PRODUCT OF DERIVATIVE ARITHMETIC WITH AN ARITHMETIC FUNCTION MULTIPLICATIVE Es-Said En-Naoui To cite this version: Es-Said En-Naoui. DIRICHLET PRODUCT OF DERIVATIVE ARITHMETIC WITH AN ARITH- METIC FUNCTION MULTIPLICATIVE. 2019. hal-02267882 HAL Id: hal-02267882 https://hal.archives-ouvertes.fr/hal-02267882 Preprint submitted on 22 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Copyright DIRICHLET PRODUCT OF DERIVATIVE ARITHMETIC WITH AN ARITHMETIC FUNCTION MULTIPLICATIVE A PREPRINT Es-said En-naoui [email protected] August 21, 2019 ABSTRACT 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. 1 Introduction Barbeau [1] defined the arithmetic derivative as the function δ : N → N , defined by the rules : 1. δ(p)=1 for any prime p ∈ P := {2, 3, 5, 7,...,pi,...}. 2. δ(ab)= δ(a)b + aδ(b) for any a,b ∈ N (the Leibnitz rule) . s αi Let n a positive integer , if n = i=1 pi is the prime factorization of n, then the formula for computing the arithmetic derivative of n is (see, e.g., [1, 3])Q giving by : s α α δ(n)= n i = n (1) pi p Xi=1 pXα||n A brief summary on the history of arithmetic derivative and its generalizations to other number sets can be found, e.g., in [4] . arXiv:1908.07345v1 [math.GM] 15 Aug 2019 First of all, to cultivate analytic number theory one must acquire a considerable skill for operating with arithmetic functions. We begin with a few elementary considerations. Definition 1 (arithmetic function). An arithmetic function is a function f : N −→ C with domain of definition the set of natural numbers N and range a subset of the set of complex numbers C. Definition 2 (multiplicative function). A function f is called an multiplicative function if and only if : f(nm)= f(n)f(m) (2) for every pair of coprime integers n,m. In case (2) is satisfied for every pair of integers n and m , which are not necessarily coprime, then the function f is called completely multiplicative. α1 αs Clearly , if f are a multicative function , then f(n) = f(p1 ) ...f(ps ), for any positive integer n such that n = α1 αs α1 αs p1 ...ps , and if f is completely multiplicative , so we have : f(n)= f(p1) ...f(ps) . Example 3. Let n ∈ N∗ , This is the same classical arithmetic functions used in this article : 1. Identity function : The function defined by Id(n)= n for all positive integer n. Dirichlet product of derivative arithmetic with an arithmetic function multiplicative A PREPRINT n 2. The Euler phi function : ϕ(n)= 1. k=1 gcd(Pk,n)=1 3. The number of distinct prime divisors of n : ω(n)= 1 . pP|n 1 if n =1 4. The Mobiuse function : µ(n)= 0 if p2|n for some prime p (−1)ω(n) otherwise 5. number of positive divisors of n defined by : τ(n)= 1 . dP|n 6. sum of divisors function of n defined by : σ(n)= d . dP|n Now ,if f,g : N −→ C are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution f ∗ g is a new arithmetic function defined by: n (f ∗ g)(n)= f(d)g( )= f(a)g(b) (3) d Xd|n abX=n where the sum extends over all positive divisors d of n , or equivalently over all distinct pairs (a,b) of positive integers whose product is n. In particular, we have (f ∗ g)(1) = f(1)g(1) ,(f ∗ g)(p)= f(1)g(p)+ f(p)g(1) for any prime p and for any power prime pm we have : m (f ∗ g)(pm)= f(pj)g(pm−j ) (4) Xj=0 This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients: f ∗ g (n) f(n) g(n) = (5) ns ns ns nX≥1 nX≥1 nX≥1 with Riemann zeta function or is defined by : 1 ζ(s)= ns nX≥1 These functions are widely studied in the literature (see, e.g., [5, 6, 7]). Now before to past to main result we need this propriety , if f and g are multiplicative function , then f ∗ g is multiplicative. 2 Main results In this section we give the new result of Dirichlet product between derivative arithmetic and an arithmetic function multiplicative f , and we will give the relation between τ and the derivative arithmetic . Theorem 4. Given a multiplicative function f, and lets n and m two positive integers such that gcd(n,m)=1 , Then we have : (f ∗ δ)(nm)= Id ∗ f (n). f ∗ δ (m)+ Id ∗ f (m). f ∗ δ (n) (6) 2 Dirichlet product of derivative arithmetic with an arithmetic function multiplicative A PREPRINT Proof. Lets n and m two positive integers such that gcd(n,m)=1, and let f an arithmetic function multiplicative , then we have : nm nm n m (f ∗ δ)(nm)= f δ(d)= f( )δ(d1d2)= f( )f( ) d1δ(d2)+ d2δ(d1) d d1d2 d1 d2 dX|nm dX1|n dX1|n d2|m d2|m n m m n = d1f( )f( )δ(d2)+ d2f( )f( )δ(d1) d1 d2 d2 d1 dX1|n d2|m n m m n = d1f( ) f( )δ(d2) + d2f( ) f( )δ(d1) d1 d2 d2 d1 dX1|n dX2|m dX2|m dX1|n = Id ∗ f (n). f ∗ δ (m)+ Id ∗ f (m). f ∗ δ (n) s αi Lemma 5. For any natural number n , if n = i=1 pi is the prime factorization of n, then : Q s f ∗ δ (pαi ) f ∗ δ (n)= Id ∗ f (n) i (7) Id ∗ f (pαi ) Xi=1 i α1 αs Proof. Let n a positive integer such that n = p1 ...ps and let f an arithmetic function , Then : α1 αs f ∗ δ (n)= f ∗ δ (p1 ...ps ) α2 αs α1 α1 α2 αs = Id ∗ f (p2 ...ps ). f ∗ δ (p1 )+ Id ∗ f (p1 ). f ∗ δ (p2 ...ps ) f ∗ δ (pα1 ) = Id ∗ f (n). 1 + Id ∗ f (pα1 ). Id ∗ f (pα3 ...pαs ). f ∗ δ (pα2 )+ Id ∗ f (pα1 ) 1 3 s 2 1 + Id ∗ f (pα2 ). f ∗ δ (pα3 ...pαs ) 2 3 s f ∗ δ (pα1 ) f ∗ δ (pα2 ) = Id ∗ f (n) 1 + Id ∗ f (n) 2 + Id ∗ f (pα1 ) Id ∗ f (pα2 ) 1 2 α1 α2 α3 αs + Id ∗ f (p1 ) Id ∗ f (p2 ) f ∗ δ (p3 ...ps ) . . f ∗ δ (pα1 ) f ∗ δ (pα2 ) f ∗ δ (pαs ) = Id ∗ f (n) 1 + Id ∗ f (n) 2 + ... + Id ∗ f (n) s Id ∗ f (pα1 ) Id ∗ f (pα2 ) Id ∗ f (pαs ) 1 2 s f ∗ δ (pα1 ) f ∗δ (pαs) = Id ∗ f (n) 1 + ... + s Id ∗ f (pα1 ) Id ∗ f (pαs ) 1 s s f ∗δ (pαi ) = Id ∗ f (n) i Id ∗ f (pαi ) Xi=1 i α s f δ p i s αi ∗ ( i ) an other prof by induction on s that if n = i pi then (f ∗ δ)(n)= Id ∗ f (n) α . =1 Id∗f (p i ) Q iP=1 i N s αi Proof. Consider n ∈ and express n = i=1 pi where all pi are distinct . α α 1 f∗δ (p i ) f∗δ (p 1 ) Q i α1 1 α1 where s =1 , it is clear that (f ∗ δ)(n)= Id ∗ f (n) α = Id ∗ f (p ) α = f ∗ δ (p1 ). Id∗f (p i ) Id∗f (p 1 ) iP=1 i 1 3 Dirichlet product of derivative arithmetic with an arithmetic function multiplicative A PREPRINT s αi Assume that n = i=1 pi , then we have : αs+1 Q αs+1 αs+1 id ∗ δ (n.ps+1 )= Id ∗ f (ps+1 ). f ∗ δ (n)+ Id ∗ f (n). f ∗ δ (ps+1 ) s f ∗δ (pαi ) f ∗ δ (pαs+1 ) = Id ∗ f (pαs+1 ). Id ∗ f (n) i + Id ∗ f (pαs+1 ). Id ∗ f (n) s+1 s+1 Id ∗ f (pαi ) s+1 Id ∗ f (pαs+1 ) Xi=1 i s+1 s αs+1 f ∗ δ (pαi ) f ∗ δ (p ) = Id ∗ f (n.pαs+1 ) i + Id ∗ f (n.pαs+1 ) s+1 s+1 Id ∗ f (pαi ) s+1 Id ∗ f (pαs+1 ) Xi=1 i s+1 s αs+1 f ∗ δ (pαi ) f ∗ δ (p ) = Id ∗ f (n.pαs+1 ) i + s+1 s+1 Id ∗ f (pαi ) Id ∗ f (pαs+1 ) Xi=1 i s+1 s+1 f ∗ δ (pαi ) = Id ∗ f (n.pαs+1 ) i s+1 Id ∗ f (pαi ) Xi=1 i Proposition 6. Let f a function arithmetic multiplicative , and δ the derivative arithmetic , then we have : 1 Id ∗ δ (n)= τ(n)δ(n) (8) 2 n Proof. Since (Id ∗ Id)(n)= d d = n 1= nτ(n). d|n d|n α P P α α j α−j j−1 α−j 1 α−1 and : (Id ∗ δ)(p )= δ(p )Id(p )= jp p = 2 α(α + 1)p .
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