Class Of Arithmetic Functions Research Articles

An identity for a class of arithmetical functions of several variables

Johnson [1] evaluated the sum∑d|n|C(d;r)|, whereC(n;r)denotes Ramanujan's trigonometric sum. This evaluation has been generalized to a wide class of arithmetical functions of two variables. In this paper, we generalize this evaluation to a wide class of arithmetical functions of several variables and deduce as special cases the previous evaluations.

An identity for a class of arithmetical functions of two variables

For a positive integer r, let r∗ denote the quotient of r by its largest squarefree divisor (1∗ = 1). Recently, K. R. Johnson proved that urn:x-wiley:01611712:media:ijmm180307:ijmm180307-math-0001 according as r∗|n or not where C(n, r) is the well known Ramanujan′s sum. In this paper, using a different method, we generalize (∗) to a wide class of arithmetical functions of 2 variables and deduce as special cases (∗) and similar formulae for several generalizations of Ramanujan′′s sum.

On some classes of arithmetical functions on a semigroupG K

On some classes of arithmetical functions on a semigroupG K

Local limit distribution laws for a class of arithmetic functions

Local limit distribution laws for a class of arithmetic functions

Identities involving the coefficients of a class of Dirichlet series. VII

Let a ( n ) a(n) be an arithmetical function, and consider the Riesz sum A ρ ( x ) = Σ n ≤ x a ( n ) ( x − n ) ρ {A_\rho }(x) = {\Sigma _{n \leq x}}a(n){(x - n)^\rho } . For a ( n ) a(n) belonging to a certain class of arithmetical functions, A ρ ( x ) {A_\rho }(x) can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of ρ \rho . Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.

Identities Involving the Coefficients of a Class of Dirichlet Series. VII

Let $a(n)$ be an arithmetical function, and consider the Riesz sum ${A_\rho }(x) = {\Sigma _{n \leq x}}a(n){(x - n)^\rho }$. For $a(n)$ belonging to a certain class of arithmetical functions, ${A_\rho }(x)$ can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of $\rho$. Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.

A New Method in Arithmetical Functions and Contour Integration

If f is a suitable meromorphic function, then by a classical technique in the calculus of residues, one can evaluate in closed form series of the form,Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration(1.1)where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.

On a class of arithmetical functions

On a class of arithmetical functions

Certain classes of arithmetic functions and the operation of additive convolution.

Certain classes of arithmetic functions and the operation of additive convolution.

A Congruence for a Class of Arithmetic Functions

There is considerable literature concerning the century old result that for arbitrary positive integers a and m, 1.1 where μ(m) is the usual Mobius function. For earlier work on this we refer to L.E. Dickson [4, pp. 84–86] and L. Carlitz [1,2]. Another reference not noted by the above authors is R. Vaidyanathaswamy [6], who noted that the left member of (1.1) represents the number of special fixed points of the m th power of a rational transformation of the n th degree.

A general class of arithmetical functions

A general class of arithmetical functions

Generating Functions for a Class of Arithmetic Functions

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

Generating Functions for a Class of Arithmetical Functions

Generating Functions for a Class of Arithmetical Functions

A Generating Function for a Class of Arithmetic Functions

(1963). A Generating Function for a Class of Arithmetic Functions. The American Mathematical Monthly: Vol. 70, No. 8, pp. 841-842.

A Generating Function for a Class of Arithmetic Functions

A Generating Function for a Class of Arithmetic Functions

Note on a Class of Arithmetical Functions

Note on a Class of Arithmetical Functions

Averages of completely even arithmetical functions

For certain classes of arithmetical functions f(n), approximations are obtained for the sum ∑f((a, b)), where {a, b} is permitted to range over plane regions of traingular, circular, and hyperbolic type.

A Class of Arithmetical Functions in Several Variables With Applications to Congruences

A Class of Arithmetical Functions in Several Variables With Applications to Congruences

A class of arithmetical functions in several variables with applications to congruences

A class of arithmetical functions in several variables with applications to congruences

A CLASS OF ARITHMETICAL FUNCTIONS

A CLASS OF ARITHMETICAL FUNCTIONS