OOn a New Formula for Arithmetic Functions
J.AKOUNSeptember 15, 2020
Abstract
In this paper we establish a new formula for the arithmetic functions that verify f ( n ) = (cid:80) d | n g ( d ) where g is also an arithmetic function. We prove the followingidentity, ∀ n ∈ N ∗ , f ( n ) = n (cid:88) k =1 µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (cid:98) nk (cid:99) (cid:88) l =1 g ( kl ) kl where ϕ and µ are respectively Euler’s and Mobius’ functions and (.,.) is the GCD.First, we will compare this expression with other known expressions for arithmeticfunctions and pinpoint its advantages. Then, we will prove the identity usingexponential sums’ proprieties. Finally we will present some applications with wellknown functions such as d and σ which are respectively the number of divisorsfunction and the sum of divisors function. In the paper we present and prove a new transformed expression for some arithmeticfunctions. First let’s remind the definition of an arithmetic function,
Definition.
A function f is arithmetic if its domain is the positive integers and hence f : N ∗ −→ C . The main theorem is,
Theorem. If f ( n ) = (cid:80) d | n g ( d ) where g is an arithmetic function then, ∀ n ∈ N ∗ , f ( n ) = n (cid:88) k =1 µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (cid:98) nk (cid:99) (cid:88) l =1 g ( kl ) kl where ϕ and µ are respectively Euler’s and Mobius’ functions and (.,.) is the GCD. The strength of this new expression is that the sum is not indexed on divisors but on all1 a r X i v : . [ m a t h . G M ] S e p he integers. As a matter of fact let’s take the example of σ which is the sum of divisorsfunction. We can clearly write σ as a sum with, σ ( n ) = (cid:88) k | n k however this formula is not really interesting because we don’t control the indexation.With this expression we moved the unknown from σ to the indexation, whereas our trans-formed expression for arithmetic functions does not hide complexity in the indexation.For σ the theorem gives us, ∀ n ∈ N ∗ , σ ( n ) = n (cid:88) k =1 (cid:106) nk (cid:107) µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) Still, the uncontrolled arithmetical part has not completely vanished and lies in Euler’sand Mobius’ functions.The idea to express arithmetical functions in terms of sums was first explored byRamanujan’s who gave expressions with series. For example he proved that, ∀ n ∈ N ∗ , σ ( n ) = π ∞ (cid:88) k =1 nk µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) Our expression is reminiscent of Ramanujan’s one. This will turn out coherent since theproof of our theorem relies on Ramanujan’s sums. Nevertheless let’s underline that ourtheorem gives expressions with finite sums that might be easier to manipulate than series.Finally, finding a new formulas for arithmetical functions is always exciting as theyare at the core of great modern problems. For instance, σ is closely linked to the Riemannhypothesis (RH) as underline the two following equivalences. Robin’s equivalence RH ⇐⇒ ∀ n ≥ , σ ( n ) < ne γ log ( log ( n ))where γ is Euler-Mascheroni constant. Lagarias’ equivalence RH ⇐⇒ ∀ n > , σ ( n ) < H n + log ( H n ) e H n where H n = (cid:80) nk =1 1 k . 2 Preliminary results on exponential sums
First we will focus on classical exponential sums. Those sums are well known andwill be useful to establish the theorem presented in this article. For those sums we knowthat, ∀ n ∈ N ∗ , ∀ m ∈ N , n (cid:88) k =1 exp (cid:18) iπkmn (cid:19) = (cid:40) n if n | m otherwise (1)This result can easily be established with the formula of geometric sums. The second type of exponential sums which will turn out useful for the proof areRamanujan’s sums. They are defined by, c m ( n ) = (cid:88) ≤ k ≤ n ( k,m )=1 exp (cid:18) iπknm (cid:19) Those sums are harder to study than the ones before. Still we can remark that,If ( m, n ) = 1, Φ = (cid:40) Z /m Z → Z /m Z x (cid:55)−→ n · x is an isomorphism. Thus, c m ( n ) = (cid:88) ≤ k ≤ m ( k,m )=1 exp (cid:18) iπknm (cid:19) = (cid:88) ≤ k ≤ m ( k,m )=1 exp (cid:18) iπkm (cid:19) = c m (1)moreover we know that, (cid:88) d | n µ ( d ) = (cid:40) if n = 10 otherwise (2)if we rearrange the sum it comes, ∀ m ∈ N ∗ , c m (1) = (cid:88) ≤ k ≤ m ( k,m )=1 exp (cid:18) iπkm (cid:19) = m (cid:88) k =1 exp (cid:18) iπkm (cid:19) (cid:88) d | ( k,m ) µ ( d )= (cid:88) d | m µ ( d ) md (cid:88) l =1 exp (cid:18) iπlm (cid:19) (cid:88) d | m µ ( d ) · d = m = µ ( m )If ( m, n ) = n , c m ( n ) = (cid:88) ≤ k ≤ n ( k,m )=1 exp (cid:18) iπkmn (cid:19) = (cid:88) ≤ k ≤ n ( k,m )=1 ϕ ( m ) Remark.
We already see with those two examples that Ramanujan’s sums are closelylinked with Euler’s and Mobius’ functions.Except for those values of ( m, n ) the expression of c m ( n ) is harder to find. FortunatelyHlder showed in 1936 that, ∀ m ∈ N ∗ , ∀ n ∈ N , c m ( n ) = µ (cid:18) m ( m, n ) (cid:19) ϕ ( m ) ϕ (cid:16) m ( m,n ) (cid:17) (3) Let f be an arithmetic function such as f ( n ) = (cid:80) d | n g ( d ) where g is also an arith-metic function. Thanks to the equality (1) we can write f as a sum, ∀ n ∈ N ∗ , f ( n ) = (cid:88) k | n g ( k ) = n (cid:88) k =1 g ( k ) k k (cid:88) l =1 exp (cid:18) iπnlk (cid:19) (4)Let’s rewrite this sum by changing the indexation, f ( n ) = (cid:88) ≤ a ≤ b ≤ n ( a,b )=1 exp (cid:18) iπnab (cid:19) C (cid:16) ab (cid:17) (5)where, C (cid:16) ab (cid:17) = (cid:88) k ∈ E ( ab ) g ( k ) k (6)with, E (cid:16) ab (cid:17) = (cid:110) ≤ k ≤ n, ∃ l ≤ k, lk = ab (cid:111) = (cid:110) ≤ k ≤ n, ∃ l ≤ k, ∃ u ∈ N ∗ | ( k, l ) = ( ub, ua ) (cid:111) = (cid:110) k = ub f or ≤ u ≤ (cid:106) nb (cid:107) (cid:111) C (cid:16) ab (cid:17) = (cid:98) nb (cid:99) (cid:88) u =1 g ( ub ) ub (7)When we use the equality (7) in (5) it comes, ∀ n ∈ N ∗ , f ( n ) = (cid:88) ≤ a ≤ b ≤ n ( a,b )=1 exp (cid:18) iπnab (cid:19) (cid:98) nb (cid:99) (cid:88) u =1 g ( bu ) bu = n (cid:88) b =1 (cid:98) nb (cid:99) (cid:88) u =1 g ( bu ) bu n (cid:88) a =1( a,b )=1 exp (cid:18) iπnab (cid:19) = n (cid:88) b =1 c b ( n ) (cid:98) nb (cid:99) (cid:88) u =1 g ( bu ) bu (8)where c b ( n ) is the Ramanujan’s sum defined in the preliminaries.Finally, when we replace with the expression (2) given in the preliminaries we have, ∀ n ∈ N ∗ , f ( n ) = n (cid:88) k =1 µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (cid:98) nk (cid:99) (cid:88) l =1 g ( kl ) kl (9)which is the identity we wanted to prove. Here we study the function σ γ ( n ) = (cid:80) d | n d γ = (cid:80) d | n g ( d ) where g ( d ) = d γ . Thisis a generalization of the sum of divisors function ( γ = 1) and the number of divisorsfunction ( γ = 0). When we apply our theorem it comes, ∀ n ∈ N ∗ , σ γ ( n ) = n (cid:88) k =1 c k ( n ) k γ − (cid:98) nk (cid:99) (cid:88) l =1 l γ − = n (cid:88) k =1 µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) k γ − (cid:98) nk (cid:99) (cid:88) l =1 l γ − (10)Let’s remark that this expression is true for γ ∈ R and not only for integers.5or γ = 0 in (10) we have, ∀ n ∈ N ∗ , d ( n ) = n (cid:88) k =1 c k ( n ) k H (cid:98) nk (cid:99) = n (cid:88) k =1 k H (cid:98) nk (cid:99) µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (11)where H n = (cid:80) ni =1 1 i for n ∈ N ∗ .For γ = 1 in (10) we have, ∀ n ∈ N ∗ , σ ( n ) = n (cid:88) k =1 c k ( n ) (cid:106) nk (cid:107) = n (cid:88) k =1 (cid:106) nk (cid:107) µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (12) Let’s call Π( n ) = (cid:81) d | n d . Then, we have log (Π( n )) = (cid:80) d | n log ( d ). We can applythe theorem and it comes, ∀ n ∈ N ∗ , log (Π( n )) = n (cid:88) k =1 µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (cid:98) nk (cid:99) (cid:88) l =1 log ( kl ) kl (13)Moreover, ∀ n ∈ N ∗ , Π ( n ) = (cid:89) d | n d · (cid:89) d | n nd = (cid:89) d | n n = n d ( n ) thus, ∀ n ∈ N ∗ , log (Π( n )) = 12 d ( n ) · log ( n ) (14)when we equalize (14) and (13) with equality (11) it comes, ∀ n ∈ N ∗ , n (cid:88) k =1 µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (cid:98) nk (cid:99) (cid:88) l =1 log (cid:16) ( kl ) n (cid:17) kl = 0 (15) Lets define Kronecker function as δ ( n ) = (cid:80) d | n µ ( d ). According to (2) we know that, δ ( n ) = (cid:40) if n = 10 otherwise n (cid:88) k =1 µ (cid:18) k ( n, k ) (cid:19) ϕ ( k ) ϕ (cid:16) k ( n,k ) (cid:17) (cid:98) nk (cid:99) (cid:88) l =1 µ ( kl ) kl = (cid:40) if n = 10 otherwise (16) References [1] A. F. Mobius, ber eine besondere Art von Umkehrung der Reihen . Journal fr die reineund angewandte Mathematik, 1832.[2] S. Ramanujan,
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