Please do the second and third one thanks! 1. Prove or provide a counterexample to each of the following statements:• If f and g are completely multiplicative arithmetic functions, then f *g must alsobe completely multiplicative.Proof:* gcd(m, n). Any divisor of mn is the form d = d,d, where d | m and d, | n* m, n has no shared divisor.. f * g(mn) = Eamn f (d)g() = di|m Edz]n f(d1d2)g()= 1тп: Σm Σn f(di)f(d)g(Ή)g(X). Σf(d1)g(H) Σιμ f(d)g( )dı|md2|n:: (f * g)(m)(f * g)(n)• If f and g are arithmetic functions and f * g is completely multiplicative, thenboth f and g must be completely multiplicative.• Convolution of arithmetic functions is associative: (f * g) * h = f * (g * h) forarithmetic functions f, g and h.

• If f and g are completely multiplicative arithmetic functions, then f *g must also be completely multiplicative. Proof: : дcd(m, п) .. Any divisor of mn is the form d = d,d2 where d1 | m and d, | n * m, n has no shared divisor .. f * g(mn) = Ea edmn (d)g() = Edilm Edzln F(d1d2)g() Σ Σm f(di)f(da)g(Η )g(H) . Σmf (d1)g(H) Σ.u f (d%) g ( H) :: (f * g)(m)(f * g)(n) • If f and g are arithmetic functions and f * g is completely multiplicative, then dı|m 2|n both f and g must be completely multiplicative. • Convolution of arithmetic functions is associative: (f * g) * h = f * (g * h) for arithmetic functions f, g and h.