Prime number theorem | Wikipedia audio article

This is an audio version of the Wikipedia Article:


00:01:48 1 Statement
00:05:51 2 History of the proof of the asymptotic law of prime numbers
00:06:29 3 Proof sketch
00:09:52 4 Prime-counting function in terms of the logarithmic integral
00:10:50 5 Elementary proofs
00:24:30 6 Computer verifications
00:26:10 7 Prime number theorem for arithmetic progressions
00:33:33 7.1 Prime number race
00:33:50 8 Non-asymptotic bounds on the prime-counting function
00:34:03 9 Approximations for the ispann
00:38:08 10 Table of spaniπ/i(ix/i)
00:39:17 11 Analogue for irreducible polynomials over a finite field
00:40:28 12 See also
00:40:52 13 Notes
00:42:50 14 References
00:45:01 15 External links
00:48:18 Table of π(x), x / log x, and li(x)
00:48:57 Analogue for irreducible polynomials over a finite field
00:53:08 n, and it is not difficult to bound the remaining terms. The "Riemann hypothesis" statement depends on the fact that the largest proper divisor of n can be no larger than n/2.
00:53:25 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
Other Wikipedia audio articles at:
Upload your own Wikipedia articles through:
Speaking Rate: 0.7686267807713529
Voice name: en-GB-Wavenet-B


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).
The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N).