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Does every digit occur with equal frequency in the set of prime numbers?

More precisely, let $f(n)$ be the total number of base-$b$ digits contained in the first $n$ prime numbers and let $f_d(n)$ be the number of times the digit $d$ occurs in the first $n$ prime numbers. Is it true that

$$\lim_{n\rightarrow \infty}\frac{f_d(n)}{f(n)}=\frac{1}{b}$$

for every base $b$ and every base-$b$ digit $d$?

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Copeland and Erdős showed that the concatenated (in base $b$) digits of the primes (and other sufficiently-dense sequences) are normal in that base, and so every digit occurs with equal frequency in the primes.

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