ADDITIVE TRANSFORM OF AN ARITHMETIC FUNCTION : PART I (original) (raw)

A note on generating identities for multiplicative arithmetic functions

Contributions to mathematics, 2023

In this article, it is shown that many of the identities involving multiplicative arithmetic functions are special cases of a more general formula. The approach employed in this article avoids using classical techniques, including Dirichlet's convolution.

Various Arithmetic Functions and their Applications

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, University of Craiova Library, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, România). The book is based on various articles in the theory of numbers (starting from 1975), updated many times.

Summatory Multiplicative Arithmetic Functions: Scaling andRenormalization

2011

We consider a wide class of summatory functions F {f ; N, p m } = k≤N f (p m k), m ∈ Z + ∪ {0}, associated with the multiplicative arithmetic functions f of a scaled variable k ∈ Z + , where p is a prime number. Assuming an asymptotic behavior of summatory function, F {f ; N, 1} N →∞ = G 1 (N) [1 + O (G 2 (N))], where G 1 (N) = N a1 (log N) b1 , G 2 (N) = N −a2 (log N) −b2 and a 1 , a 2 ≥ 0, −∞ < b 1 , b 2 < ∞, we calculate a renormalization function defined as a ratio, R (f ; N, p m) = F {f ; N, p m } /F {f ; N, 1}, and find its asymptotics R ∞ (f ; p m) when N → ∞. We prove that the renormalization function is multiplicative, i.e., R ∞ (f ; n i=1 p mi i) = n i=1 R ∞ (f ; p mi i) with n distinct primes p i. We extend these results on the others summatory functions k≤N f (p m k l), m, l, k ∈ Z + and k≤N n i=1 f i (kp mi), f i = f j , m i = m j. We apply the derived formulas to a large number of basic summatory functions including the Euler φ(k) and Dedekind ψ(k) totient functions, divisor σ n (k) and prime divisor β(k) functions, the Ramanujan sum C q (n) and Ramanujan τ Dirichlet series, and others.

Alternative Representations of Some Arithmetic Functions

PROOF

This article presents some results of the attempt to simplify the writing of arithmetic functions on the computer so that users can apply them without additional operations, such as summing after a set whose elements must be calculated, such as the set of numbers prime. The important role of the remainder function in defining most arithmetic functions is highlighted. Defining algorithms for the prime factorization of natural numbers highlights the possibility of representing natural numbers in a basis as "natural" as possible for natural numbers, namely the basis of prime numbers. The disadvantage of this natural basis is, for the time being, that it is infinitely dimensional. For now, this representation provides advantages but also disadvantages. Among the arithmetic functions proposed in the article, there are also statistical characterizations of the distribution of prime numbers, given with the hope of helping a better knowledge of the set of prime numbers. The import...

On the binomial convolution of arithmetical functions

2008

Let n = p p νp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f • g)(n) = d|n p νp(n) νp(d) f (d)g(n/d), where a b is the binomial coefficient. We provide properties of the binomial convolution. We study the C-algebra (A, +, •, C), characterizations of completely multiplicative functions, Selberg multiplicative functions, exponential Dirichlet series, exponential generating functions and a generalized binomial convolution leading to various Möbius-type inversion formulas. Throughout the paper we compare our results with those of the Dirichlet convolution *. Our main result is that (A, +, •, C) is isomorphic to (A, +, * , C). We also obtain a "multiplicative" version of the multinomial theorem.

A study of a curious arithmetic function

Journal of Integer Sequences, 2010

In this note, we study the arithmetic function f : Z * + → Q * + defined by f (2 k ℓ) = ℓ 1−k (∀k, ℓ ∈ N, ℓ odd). We show several important properties about that function and then we use them to obtain some curious results involving the 2-adic valuation.

ON THE EVALUATION OF CERTAIN ARITHMETICAL FUNCTIONS OF NUMBER THEORY AND THEIR SUMS

In this paper we present a method to get the prime counting function π(x) and other arithmetical functions than can be generated by a Dirichlet series, first we use the general variational method to derive the solution for a Fredholm Integral equation of first kind with symmetric Kernel K(x,y)=K(y,x), after that we find another integral equations with Kernels K(s,t)=K(t,s) for the Prime counting function and other arithmetical functions generated by Dirichlet series, then we could find a solution for π(x) and () () n x a n A x ≤ = ∑ , solving [ ] 0 J δ φ = for a given functional J, so the problem of finding a formula for the density of primes on the interval [2,x], or the calculation of the coefficients for a given arithmetical function a(n), can be viewed as some " Optimization " problems that can be attacked by either iterative or Numerical methods (as an example we introduce Rayleigh-Ritz and Newton methods with a brief description) Also we have introduced some conjectures about the asymptotic behavior of the series () n n p x x p ≤ Ξ = ∑ =S n (x) for n>0 , and a new expression for the Prime counting function in terms of the Non-trivial zeros of Riemann Zeta and its connection to Riemman Hypothesis and operator theory.