A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by placing a decimal point and writing the base $n$ representation of the sequence of natural numbers in the reverse order. Read more.

This question is motivated by curiosity to study the analogue of van der Corput sequence for prime numbers. Consider the sequence $v_p$ formed by placing a decimal point and writing the digits of the sequence of prime numbers $p$ in the reverse order (in base 10). The first few terms of this sequence are

$$ 0.2, 0.3, 0.5, 0.7, 0.11, 0.31, 0.71, 0.91, 0.32, 0.92, \ldots $$

Clearly $v_p$ is not equidistributed in the unit interval and therefore it is not a low discrepancy sequence. I am curious to know if $v_p$ has anything interesting property in it.

Q1. What is the mean value of the sequence $v_p$? In other words does the following limit exist?

$$ \lim_{x \rightarrow \infty}\frac{1}{x}\sum_{p \le x}v_p. $$

**Edit** (*Adding Timothy's guess as a question*)

Q2. What would be the mean value of base $b$?

Q3. **Edit** (*@ one more variation for this problem*)

Find a closed form representation of

$$ \lim_{x\to\infty}\frac{1}{x}\sum_{p\le x}f(v_p) $$

where $f$ is any function Riemann integrable in $(0,1)$?

Considering the analogy with the equidistribution theorem, I think this should be possible.

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