# Exclusive prime factors

Let $S$ be an finite or infinite subset of the primes. Let $f(x)=1$ if $x$ has no factors in $S$. If not, $f(x)=0$. Is there a way to calculate the limit $\displaystyle\sum_{n=1}^{x} f(n)/x$, as $x$ grows big?

Examples;
S contains all primes of form $2778t+79$
S={2,5,17}

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 You should clarify "$2778t + 79$." What are the restrictions on $t$? Furthermore, can you show there are an infinity of primes of the form $2778t + 79$? – ThisIsNotAnId Jun 4 '12 at 0:01 Since $\mathrm{gcd}(2778,79)=1$, then by Dirichlet theorem (en.wikipedia.org/wiki/…) this progression contains infinitely many primes – Norbert Jun 4 '12 at 0:04 For finite set i think its easy, because f(n) is periodic, right? – Sigma23 Jun 4 '12 at 0:06

Yes, for a finite set it's easy. For your example, $S=\{{2,5,17\}}$, $x$ has no factors in $S$ if and only if $x$ is relatively prime to $170=2\times5\times17$. The number of integers less than and relatively prime to 170 is $\phi(170)=1\times4\times16=64$, and then it's periodic beyond 170, so your limit is $64/170$, also known as $32/85$.
EDIT: A well-known example. The numbers that can be written as a sum of two squares, that's known to be a set of density zero (see, e.g., http://www.math.niu.edu/~rusin/known-math/93_back/2squares). Now those include all the numbers that have no prime factors of the form $4t+3$, so it follows that the set of numbers with no prime factor $4t+3$ has density zero.