Gauss guessed
that the probability that a number N is prime
is 1/log(N) where log is taken to the base e.
This is the probability that a die with log(N)
sides lands on the "PRIME" side. Notice
that as N gets bigger, log(N) gets bigger and
the chance of landing on the prime side gets smaller.
Primes get rarer as we count higher.
If Nature tosses the prime number dice 100,000
times, how many primes will you expect to get
with these dice with varying numbers of sides?
If the die has a fixed number of sides, say 6,
then you expect 100,000/6 which is the probability
1/6 added up 100,000 times. Now Gauss is varying
the number of sides on the die at each throw.
The resulting number of primes is expected to
be:
1/log(2)+1/log(3)+…+1/log(100,000)
Gauss refined this guess at the number of primes
into a function called the logarithmic integral,
denoted by Li(N). How good is Gauss's guess compared
to the real number of primes?
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