Closed forms for $\lim_{x\rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ Im looking for closed forms for $\lim_{x \rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ where $x$ is a positive real, $a$ is a given real, $p$ is the set of primes such that the inequation is valid. Lets call this Limit $L(a)$. Mertens gave a closed form for $L(0)$. Are there others possible ?
Also the inverse function $L^{-1}(b)=a$ intrests me. What is the value of $L^{-1}(0)$ ??
 A: I always take the log when I
run into a product,
so,
pressing on regardless,
 let's look at
$\begin{align}
f(x, a)
&=\sum_{x>(p-a)>0}\ln(1-(p-a)^{-1})\\
&=\sum_{a < p < x+a}\ln(1-(p-a)^{-1})\\
&=\sum_{a < p < x+a}-(\frac1{p-a}+\frac1{2(p-a)^2}+...)\\
&=-\sum_{a < p < x+a}\frac1{p-a}+C\\
&=C-\sum_{a < p < x+a}\frac1{p(1-a/p)}\\
&=C-\sum_{a < p < x+a}\frac1{p}(1+a/p+(a/p)^2+...)\\
&=C-\sum_{a < p < x+a}\frac1{p}+\sum_{a < p < x+a}\frac{a}{p^2}
+\sum_{a < p < x+a}\frac{a^2}{p^3}
+...\\
&=C_1-\sum_{a < p < x+a}\frac1{p}
\end{align}
$
where I have blithely absorbed the various convergent sums
($ \sum_{a < p < x+a}1/(p-a)^k$ and
$\sum_{a < p < x+a}1/p^k$
for $k \ge 2$)
into constants $C$ and $C_1$
which will depend on $a$.
Using the known estimate
$\sum_{p < x} \frac1{p}
\approx \ln \ln x$,
$f(x, a)
\approx C_1-\ln \ln(x+a)+\ln\ln a
\approx C_2-\ln \ln(x+a)
$.
Since
$\ln(x+a)
= \ln x + \ln(1+a/x)
\approx \ln x+a/x
$,
$\begin{align}
\ln\ln(x+a)
&\approx \ln(\ln x+a/x)\\
&= \ln(\ln x(1+a/(x \ln x))\\
&= \ln\ln x+ \ln(1+a/(x \ln x))\\
&\approx \ln\ln x+ a/(x \ln x)\\
\end{align}
$
so
$f(x, a)
\approx C_2-\ln \ln(x+a)
\approx C_2-\ln\ln x- a/(x \ln x)
$.
So,
$\ln x e^{f(x, a)}
\approx e^{\ln \ln x + C_2-\ln\ln x}
\approx C_3
$.
All this shows is that
the limit approaches a constant,
without giving much help in
evaluating the constant.
