... P_n =~ 98804779450787677801, by Mertens theorem DavidMessage 1 of 4 , Dec 3 12:46 AMView Source--- In firstname.lastname@example.org
> Calculate (P_n) at 82P_n =~ 98804779450787677801, by Mertens' theorem
Hello David. You do not understand the problem, hence your error. Need from the formulas for calculating the number of prime numbers, go to the formulas of theMessage 2 of 4 , Dec 9 10:58 PMView SourceHello David.
You do not understand the problem, hence your error.
Need from the formulas for calculating the number of prime numbers, go to the formulas of the average gap between primes. And then:
You can not know what formula is the average gap for the interval [0, P_n ^ 2] does not work for the interval [P_n ^ 2, P_ (n 1) ^ 2]
Here, the arithmetic operations (addition, subtraction) - not applicable
For example: You have an average gap for the interval [0, x]. I mean a gap in the interval [m, x]. These two middle gap can not be compared. They are incompatible. You're trying hard to prove, based on its average gap that my average gap - is wrong.
Even the preliminary calculations for the interval [P_n ^ 2, P_ (n 1) ^ 2] show that the error increases, but at certain points it returns to the minimum error.
Why are your conclusions about the infinite growth of error does not show it back to the minimal error??
In this direction, so many possibilities, you are a priori declared all wrong.
... Application of the prime number theorem to determine the average gap between primes in the interval [p[n]^2, p[n+1]^2] is valid, for large values of theMessage 3 of 4 , Dec 10 9:42 AMView Source--- In email@example.com,
> You're trying hard to prove, based on its average gap thatApplication of the prime number theorem to determine the
> my average gap is wrong.
average gap between primes in the interval [p[n]^2, p[n+1]^2]
is valid, for large values of the n-th prime p[n], since
the numbers in this range differ little in their relative size.
The average gap is asymptotic to g = log(p[n]) + log(p[n+1])
> Even the preliminary calculations for the intervalYou have fallen into a trap:
> [P_n ^ 2, P_ (n 1) ^ 2] show that the error increases,
> but at certain points it returns to the minimum error.
"Hasty generalization, a logical fallacy also known as
'the law of small numbers': the tendency for an initial
segment of data to show some bias that drops out later."
You have blinded yourself by looking only at tiny primes.
Asymptotically, your estimate is hopelessly wrong.
In the range [p[n]^2, p[n+1]^2] between the squares of the
successive primes p[n] = 10^1000 - 1769 and
p[n+1] = 10^1000 + 453 the average gap between primes is
well estimated by the prime number theorem as
g =~ 2000*log(10) =~ 4605.
Your faulty estimate would be smaller, namely
exp(Euler)*1000*log(10) =~ 4101.
Hence you would overestimate the quantity of primes Q,
in this range, by the huge error that I computed for you:
> E =~ 2*exp(-Euler)*Q - Q =~ 1.19*10^999Please do not blind yourself by looking only at tiny primes.