ON THE DIVERGENCE OF THE SMARANDACHE HARMONIC SERIES Florian Luca
n
For any positive integer n let S(n) be the minimal positive integer m such that In [3], the authors showed that
i m!.
(1)
is divergent and attempted, with limited succes, to gain information about the behaviour of the partial sum A(x)
=L
1
S(n)2
n::;x
by comparing it with both log x and log x + log log x. In this note we show that none of these two functions is a suitable candidate for the order of magnitude of A(x). Here is the result. For any 5 > 0 and x ;::: 1 denote by A,,(x) =
L
1 S(n)<i'
(2)
n::;x
B,,(x) = 10gA,,(x). Then. Theorem. For any 5 > 0,
B ()
"'x
I 2 log x > og . log log x
0
(log x ) . log log x
(4)
What the above theorem basically says is that for fixed 5 and for arbitrary E > O. there exists some constant C (depending on both 5 and E). such that A,,(x) > 2 (l - f )~ loglogz
for x > C.
(5)
Notice that equation (5) asserts that Ao (x) grows much faster than any polynomial in log(x). so one certainly shouldn't try to approximate it by a linear in log x. The Proof. In [1 Lwe showed that (6)
diverges for all 5 > O. Since the argument employed in the proof is relevant for our purposes, we reproduce it here. Let t ;::: 1 be an integer and PI < P2 < ... < Pt be the first t prime numbers. \"otice that any integer n = Ptm where m is squarefree and all the prime factors of
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