Let f : Nn → C be an arithmetic function of n variables, where n ≥ 2. We study the mean-value M(f) of f that is defined to be lim x1,...,xn→∞ 1 x1 · · · xn ! m1≤x1, ... , mn≤xn f(m1, . . . , mn), if this limit exists. We first generalize the Wintner theorem and then consider the multiplicative case by expressing the mean-value as an infinite product over all prime numbers. In addition, we study the mean-value of a function of the form (m1, m2, . . . , mn) #→ g(gcd(m1, m2, . . . , mn)), where g is a multiplicative function of one variable, and express the mean-value by the Riemann zeta function.
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