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Multiplicative with a(2) = 1, a(2^2) = -2, and a(2^e) = 0 for e >= 3, a(3) = 2, a(3^2) = -3, and a(3^e) = 0 for e >= 3, and for a prime p >= 5, a(p) = -1, and a(p^e) = 0 for e >= 2.
T. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
From Amiram Eldar, Jan 21 2024: (Start)
Multiplicative with a(2) = 1, a(2^2) = -2, and a(2^e) = 0 for e >= 3, a(3) = 2, a(3^2) = -3, and a(3^e) = 0 for e >= 3, and for a prime p >=5, a(p) = -1, and a(p^e) = 0 for e >= 2.
Sum_{k=1..n} abs(a(k)) ~ (12/Pi^2) * n. (End)
f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := Switch[e, 1, 1, 2, -2, _, 0]; f[3, e_] := Switch[e, 1, 2, 2, -3, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 21 2024 *)
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Peter H. van der Kamp, <a href="http://emis.impa.br/EMIS/journals/INTEGERS/papers/n24/n24.Abstract.html">On the Fourier transform of the greatest common divisor</a>, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
Peter H. van der Kamp, <a href="http://emis.impa.br/EMIS/journals/INTEGERS/papers/n24/n24.Abstract.html">On the Fourier transform of the greatest common divisor</a>, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
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