Prime Reciprocal Series with Blackpenredpen (Oxford Maths Interview Question)

Steve from blackpenredpen answers a real Oxford University maths admissions interview question set by Oxford Mathematician (and interviewing tutor) Dr Tom Crawford. The question looks at the divergence of the sum of the reciprocals of the prime numbers, using the Fundamental Theorem of Arithmetic, the divergence of the Harmonic Series (1/n), and the Basel Problem (sum of 1/n^{2} equals pi-squared over 6).

This is part 2 of the interview – you can find part 1 on Gabriel’s Horn here.

I just stumbled upon the YouTube channel and enjoy the videos. I wanted to ask why (given what’s written on the board at 16:51), we can’t pull out the Pi part outside of the summation (as it’s not dependent on K) and then simply say that this entire summation diverges as we have a sum of an infinite
number of values greater than 1 (since there are an infinite number of primes) multiplied by the pi^2/2 thing from the Basel Problem.

And, I did read some comments from the YouTube video that suggests there was something weird with the removal of the “1 +” part of 1 + 1/p as it seems like that would lead to convergence (i.e. the 1+ part is required to show the proof).

I am not a mathematician, and I’m not trying to detract from the usefulness and entertainment of the videos. I just want to learn where I’ve gone wrong in my thinking or if there was a mistake in the proof (as shown in the video).

I just stumbled upon the YouTube channel and enjoy the videos. I wanted to ask why (given what’s written on the board at 16:51), we can’t pull out the Pi part outside of the summation (as it’s not dependent on K) and then simply say that this entire summation diverges as we have a sum of an infinite

number of values greater than 1 (since there are an infinite number of primes) multiplied by the pi^2/2 thing from the Basel Problem.

And, I did read some comments from the YouTube video that suggests there was something weird with the removal of the “1 +” part of 1 + 1/p as it seems like that would lead to convergence (i.e. the 1+ part is required to show the proof).

I am not a mathematician, and I’m not trying to detract from the usefulness and entertainment of the videos. I just want to learn where I’ve gone wrong in my thinking or if there was a mistake in the proof (as shown in the video).

Thanks to anyone for any replies/comments.

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err pi^2/6 (basel problem) — typo in my message above

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