The gcd-sum function
نویسنده
چکیده
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 partial sums of the Dirichlet series at real values are derived, including estimates for error terms.
منابع مشابه
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.
متن کاملA 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.
متن کاملMean 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.
متن کاملOn the Bi-Unitary Analogues of Euler’s Arithmetical Function and the Gcd-Sum Function
We give combinatorial-type formulae for the bi-unitary analogues of Euler's arith-metical function and the gcd-sum function and prove asymptotic formulae for the latter one and for another related function.
متن کاملA Gcd-Sum Function Over Regular Integers Modulo n
We introduce a gcd-sum function involving regular integers (mod n) and prove results giving its minimal order, maximal order and average order.
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