Mean Value of r-gcd-sum and r-lcm-Sum Functions

Mean Values of Generalized gcd-sum and lcm-sum Functions

We consider a generalization of the gcd-sum function, and obtain its average order with a quasi-optimal error term. We also study the reciprocals of the gcd-sum and lcm-sum functions.

Weighted Gcd-Sum Functions

We investigate weighted gcd-sum functions, including the alternating gcd-sum function and those having as weights the binomial coefficients and values of the Gamma function. We also consider the alternating lcm-sum function.

A Survey of Gcd-Sum Functions

We survey properties of the gcd-sum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcd-sum function and for the function defined by the harmonic mean of the gcd’s.

The gcd-sum function

The gcd-sum is an arithmetic function defined as the sum of the gcd’s of the first n integers with n : g(n) = ∑n i=1(i, n). The function arises in deriving asymptotic estimates for a lattice point counting problem. The function is multiplicative, and has polynomial growth. Its Dirichlet series has a compact representation in terms of the Riemann zeta function. Asymptotic forms for values of par...

On a Sum Analogous to Dedekind Sum and Its Mean Square Value Formula