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Abstract:
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In this thesis we give some new asymptotic formulas for mean values of multiplicative functions of several variables depending on GCD and LCM of arguments.
We obtain an asymptotic formula with a power saving error term for the summation function of a family of generalized least common multiple and greatest common divisor functions of several integer variables. Xn 1,...,nk+`≤x [n1,...,nk]a (n1,...,nk)c , [nk+1,...,nk+`]b (nk+1,...,nk+`)d
=
Ck,a,c;`,b,d (a + 1)k(b + 1)`
xk(a+1)+`(b+1) + O xk(a+1)+`(b+1)−1 2+
and Xn 1,...,nk+`≤x
[n1,...,nk]a (n1,...,nk)c
,
[nk+1,...,nk+`]b (nk+1,...,nk+`)d (n1 ...nk)a(nk+1 ...nk+`)b
= Ck,a,c;`,b,d xk+` + O xk+`−1 2+ . Also we obtain an asymptotic formula with a power saving error term for the summation function of Euler phi-function evaluated at iterated and generalized least common multiples of four integer variables. Xn 1,n2,n3,n4≤x ϕ [n1,n2]a (n1,n2)c , [n3,n4]b (n3,n4)d = Ca,c;b,d (a + 1)2(b + 1)2 x2a+2b+4 + O x2a+2b+7 2+ . |