Summary

Important update at the end of this page January 26, 2004

Well before the advent of internet, one could see that many number theory researchers have been striving to study the behavior of the Prime Count Function pi(x).

Well before 1896, when Hadamard and de La Vallée Poussin demonstrated what is recognized as the Prime Number Theorem, it was closely conjectured already that the 'prime number density' at x can be approximated to 1/ln(x). The actual density around point x varies vs the interval considered say x-i/2 and x+i/2 and fluctuates irregularly around the value 1/ln(x). What I call here the actual prime number density in the interval x-i/2, x+i/2 is the prime count over this interval, divided by i, the length of this interval.

Legendre, well known for lots of other mathematical discoveries, devised a formula pretending to approach the prime count function :

pi(x) ~ x / (ln(x)-1.08366) = L(x)

(Update April 30, 2003)

This value 1.08366 often referred to as Legendre's constant was seeming the best choice for the range of x values Legendre could study at his time. Recent studies have proposed other values such as 0 (as per the Prime Number Theorem) or 1 which deserves further study and references.

The degree of success of this formula is already substantial in the sense that the deviation L(x)-pi(x) is 'relatively small' compared to the value x. That deviation grows steadily but as x grows, L(x) seems to become more and more parallel to pi(x). No speculation is known to exist about L-pi taking 'some where' a negative value.

(Update April 30, 2003)

Very recently Rusy Kolev produced a demonstration for a formula more accurate than Legendre's. This can be seen here and will be discussed in the future when this summary is expanded.

(end of update)

Now to approximate better the prime count function pi(x) one might think of integrating a density formula between 0 and x, speculating that the above density formula fluctuations would compensate.

That is where the famous 'logarithmic integral' appears :

pi(x) ~ integral(2;x;(1/ln(x))) = Li(x)

A recent school of thoughts is challenging the mysterious starting point of 2 in this formula, replacing it by zero. This seems meaningless when exploring high values of x.

The deviation Li(x)-pi(x) is a bit discouraging versus L-pi for 'small' values of x, but after a 'break-point', Li-pi becomes and stays the winner. In depth studies were carried out, with elaborate demonstrations, showing that for values of x 'sufficiently large', Li-pi takes negative values. It is far beyond the value x = 10^22, the highest for which the exact value of pi(x) is known at the the moment.

An important progress is one of the many contributions of Riemann : in his study, Riemann could show that Li(x) counts, not only the prime numbers, but also 'some fraction' of the powers of the prime numbers :

Li(x) = pi(x) + (1/2)pi(x^(1/2)) + (1/3)pi(x^(1/3)) + ...

(update february 8, 2004)

An interesting web page about this finding has been created by Jon Perry and can be accessed here .

(end of update)

Hence

pi(x) ~ R(x) = Li(x) - (1/2)Li(x^(1/2)) - ...

The general term of this sum is (µ(n)/n)Li(x^(1/n)) with µ(n) being the Möbius number.

This time the deviation R(x)-pi(x) has a much better behavior. Tables have been published up to x = 10^16 and show the deviation taking 'regularly' positive and negative values and where R(x)-pi(x) remains 'well centered', though the absolute values it takes increase steadily with x, which leads to speculate that at some 'high enough' value of x, the deviation Li-pi could indeed become negative.

The deviation R-pi lends itself to plenty of ongoing studies. Prime gaps and prime constellations are obviously elements of those.
A disappointing aspect of R(x) is that it involves knowledge of prime numbers themselves, obviously and fortunately not up to x.

The heart of the present study is to devise the approach of a formula where prime numbers are no longer involved, but where the deviation remains 'centered' around positive and negative fluctuations as with R(x).

The study proceeds actually in two steps : (1) Find a 'good' formula for the decay of the prime density in two consecutive intervals between the squares of consecutive prime numbers. (2) See how good this formula can remain when replacing those 'squares of prime numbers' by 'squares of approximated prime numbers.

To illustrate the first step, say that between 9 and 25 there are 5 primes i.e. a density of 5/16, and between 25 and 49 there are 6 primes i.e. a density of 6/24.

To illustrate the second step, one could replace the sequence 29, 31, 37, 41, 43, 47 by 29, 33, 37, 41, 45, 49 by an arithmetic progression of terms all between 25 and 49, with a step of 4 (which is the reciprocal of the prime density in this interval). Be prepared to the most frequent case where the step is no longer an integer.

Update January 26, 2004 : Before tackling those two steps, I suggest to examine the behavior of of pi(x) versus a discontinuous function using some "expected prime probability" concept. Just go here .

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