MathDB

Let

M, k>0

integers.
Let

X(M,k)

the (infinite) set of all integers that can be factored as

{p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}

where each

p_i

is not smaller than

M

and also each

e_i

is not smaller than

k

.
Let

Z(M,k,n)

the number of elements of

X(M,k)

not bigger than

n

.
Show that there are positive reals

c(M,k)

and

\beta(M,k)

such that

\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)

and find

\beta(M,k)

Problem Statement