MOMENTS OF THE SMARANDACHE
FUNCTIO~
Steven R Finch MathSoft Inc. 101 Main Street Cambridge, MA, USA 02142 sjinch@mathsojt.com
Given a positive integer n, let P(n) denote the largest prime factor of nand S(n) denote the smallest integer m such that n divides m! This paper extends earlier work [1] on the average value of the Smarandache function S(n) and is based on a recent asymptotic result [2]: I{n ~ N:P(n) < S(n)}j
= 01(
due
to
•W
~ln(N)J
I
J
for any positive integer j
Ford. We first prove:
Theorem 1.
where((x) is the Riemann zeta function. In particular, In(N) lim - N . E(S(N»
N~oo
lim
N~oo
In(N) -2-'
N
Var(S(N»
;r2
= -12 = 0.82246703 ... ((3)
=-..,- = 0.40068563 ... ~
Sketch of Proof. On one hand, L(k)
In(n)
= lim - k - ' E(P(n) n~oo
n
k
)
~
In(n) k lim - k - ' E(S(n) ) n~oc n
N ~""
The above summation, on the other hand, breaks into two parts: In(N) ( k k'1 lim ~·I L pen) + LS(n) N-+oo N \..P(n)=S(n) P(n)<S(n)
j
140
In(N)
f
N
n=!
= 15 m ---;:;:!. L..S(n)
k