# Sum of digits of a prime number in base 10

Define $$S(i)$$ as the sum of the digits of the $$i$$-th prime number $$p_i$$. Furthermore, define $$T(i) := \frac{S(i)}{p_i}.$$

Is it true that $$T(i) < T(i+1)$$ infinitely many times?

Response to comment: I don't have much progression other than assuming by contradiction that there is a number $$N$$ for which $$T(i) > T(i+1)$$ for all $$i>N$$. Then I want to show that this isn't true, my first idea was using Bertrand & Dirichlet (Bertrand for bounding $$p_i$$ and $$p_{i+1}$$ to eachother, and Dirichlet for constructing a prime of the form ending on $$00001$$ or something to lower its digit sum).

I am fairly confident this statement is true, as surely you have primes of the form $$10111101013201$$ with a very low digit sum, and the next prime being of the form $$10111101099999$$ for example, with a much higher digit sum and therefore making our statement very probable.

I created this problem myself, and context was purely random. I wanted to create an olympiad style problem with digit sum of a prime, but this problem seems too hard for an olympiad. I am assuming we need strong tools.

• Please edit to include your efforts and for context. Where did this problem come from? Why are you interested in it? Is there any reason to imagine it has a sensible answer?
– lulu
Commented Jul 27, 2023 at 13:26
• Please look at my edit. Commented Jul 27, 2023 at 15:25
• You're right, this looks like the kind of problems that need very strong tools, my guess is those tools are yet to be invented/discovered. Commented Jul 27, 2023 at 15:34
• Thank you for confirming my suspicion. Commented Jul 27, 2023 at 15:41

Katai proved in 1986 that the sum of digits of primes acts like a normal random variable in the limit with mean $$\frac92\log N$$ and variance $$\frac{33}4\log N$$ (a central limit theorem), which is what we'd expect if the digits of primes were distributed randomly. More precisely, Katai proved that for every real number $$y$$, $$\lim_{N\to\infty} \frac1N \# \biggl\{ i\le N\colon T(i) \le \frac92\log N + y \sqrt{\frac{33}4\log N} \biggr\} = \frac1{\sqrt{2\pi}} \int_{-\infty}^y e^{-t^2/2}\,dt.$$
It's not hard to check that this is incompatible with a world where $$T(i) \ge T(i+1)$$ always past some point (it implies, for example, that when $$N$$ is large, there are always primes $$p_i$$ with $$N\le i\le 2N$$ for which $$T(i) < \frac92\log N$$ and also primes in that range for which $$T(i) > \frac92\log N$$).