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Actually, the density you indicate is false, because $X^\theta$ is too large for the primes $p\leq X^\theta$ to behave sufficiently independently. For example, for $\theta=1/2$ the sifted set consists precisely of the primes in $(X^{1/2},X]$ whose density is $1/\log X$ (instead of $2e^\gamma/\log X$) by the Prime Number Theorem.

Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$. Then Theorem 3 in Section of III.6.2 of Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula $$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$ where $\omega:[1,\infty)\to[1/2,1]$ is the Buchstab function satisfying (cf. Corollary 3.1 after the theorem) $$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$ Better bounds and various refinements are also available: see Chapter III.6 of the book in more detail.

Actually, the density you indicate is false, because $X^\theta$ is too large for the primes $p\leq X^\theta$ to behave sufficiently independently. For example, for $\theta=1/2$ the sifted set consists precisely of the primes in $(X^{1/2},X]$ whose density is $1/\log X$ (instead of $2e^{-\gamma}/\log X$) by the Prime Number Theorem.

Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$. Then Theorem 3 in Section of III.6.2 of Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula $$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$ where $\omega:[1,\infty)\to[1/2,1]$ is the Buchstab function satisfying (cf. Corollary 3.1 after the theorem) $$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$ Better bounds and various refinements are also available: see Chapter III.6 of the book in more detail.

Actually, the density you indicate is false, because $X^\theta$ is too large for the primes $p\leq X^\theta$ to behave sufficiently independently.

Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$. Then Theorem 3 in Section of III.6.2 of Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula $$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$ where $\omega:[1,\infty)\to[1/2,1]$ is an explicit function satisfying (cf. Corollary 3.1 after the theorem) $$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$ Better bounds and various refinements are also available: see Chapter III.6 of the book in more detail.

Actually, the density you indicate is false, because $X^\theta$ is too large for the primes $p\leq X^\theta$ to behave sufficiently independently. For example, for $\theta=1/2$ the sifted set consists precisely of the primes in $(X^{1/2},X]$ whose density is $1/\log X$ (instead of $2e^\gamma/\log X$) by the Prime Number Theorem.

Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$. Then Theorem 3 in Section of III.6.2 of Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula $$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$ where $\omega:[1,\infty)\to[1/2,1]$ is the Buchstab function satisfying (cf. Corollary 3.1 after the theorem) $$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$ Better bounds and various refinements are also available: see Chapter III.6 of the book in more detail.

Actually, the bound you indicate is false, exactly because $X^\theta$ is too large. Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$. Then Theorem 3 in Section of III.6.2 of Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula $$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$ where $\omega:[1,\infty)\to[1/2,1]$ is an explicit function satisfying (cf. Corollary 3.1 after the theorem) $$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$ Better bounds and various refinements are also available: see Chapter III.6 of the book in more detail.

Actually, the density you indicate is false, because $X^\theta$ is too large for the primes $p\leq X^\theta$ to behave sufficiently independently. 

Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$. Then Theorem 3 in Section of III.6.2 of Tenenbaum: Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula $$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$ where $\omega:[1,\infty)\to[1/2,1]$ is an explicit function satisfying (cf. Corollary 3.1 after the theorem) $$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$ Better bounds and various refinements are also available: see Chapter III.6 of the book in more detail.