Expressions that give lower bounds on prime counting functions (Part 2)

0. References

[1]https://www.chart.co.jp/subject/sugaku/suken_tsushin/91/91-4.pdf, using Bertrand Chebyshev's theorem, Noriyoshi Nishimoto.

[2]https://isiotoyume.wixsite.com/website/post/隣接するフィボナッチ数の間に素数が存在する確率は, as it gets bigger, it seems to get closer to 76-4-. Yume Isioto

[3]https://youtu.be/YNmRhCNvBbg Lujandre Conjecture "College Math and Physics" studied in a prep school groove.

[4]https://isiotoyume.wixsite.com/website/post/n番目のフィボナッチ数f-n-を素数の個数の下限とする区間の取り方について Yume Isioto

[5]http://web.tuat.ac.jp/~nagaki/zairiki/singular2007.pdf 1 Finding deflection curves using singular functions

[6]https://ja.wikipedia.org/wiki/素数計数関数 Prime Number Calculating Functions

1. Relationship between square Fibonacci numbers and the closest Fibonacci number to it

First, let's square the Fibonacci numbers. (f(n)^2)

1, 4, 9, 25, 64, 169, 441,...

Next, we try to find the Fibonacci number (g(m)) that is closest to this. (Exclude 1.)

3, 8, 21, 55, 144, 377,...

Find the difference between the two, ε(n,m).

1, 1, 4, 9, 25, 64, ...

Then ε is a function of n, and

ε=ε(n)=(f(n-2))^2

then ε=ε(n)=(f(n-2))^2.

Then g(m) is also a function of n, and

g(m)=g(n)=f(n-1)(f(n+1)-f(n-3))

We can see that this is the case.

Furthermore, if we pay attention to the fact that

g(m)=f(2n-1)

(2n-1). 2.

2.1 Lower bound on the number of primes in the interval [f(t), f(t+2)] by Fibonacci numbers one place apart

2.1 Conjecture from Bertrand Chebyshev's Theorem [1]

Bertrand Chebyshev's theorem is explained in reference [1]. We will also use the lower bound on the number of primes in an interval that is a Fibonacci number [2], which was explained previously.

With probability, there can be only the following number of primes in [f(2n-1), f(2n)].

Similarly, with probability, there can be only the following number of primes in [f(2n),f(2n+1)].

From the above, with probability, there can be only the following number of primes in [f(2n-1),f(2n+1)].

The arrow indicates the case where n is skipped to infinity such that the ratio of adjacent Fibonacci numbers is close enough to the golden ratio. For convenience, the smallest Fibonacci number at which the ratio is close enough to the golden ratio is n_0.

2.2 Conjecture from the Lujandre Conjecture [3]

The Lujandre conjecture is explained in reference [3]. We will also use [4] for taking the interval where the nth Fibonacci number f(n) is a lower bound on the number of primes, as explained previously.

In [4], we explained that when the Lujandre conjecture is correct, there are at least f(n) primes in the interval [f(n+1)^2, f(n+2)^2].

Then, if we take n as n>n_0, then the number of primes in the interval [f(n+2)^2, f(n+3)^2] is at least the golden ratio (φ) times [f(n+1)^2, f(n+2)^2]. 3.

3. lower bound on the number of primes for x<n_0

Since we can't approximate the golden ratio, we should get it from the probability of the existence of prime numbers.

Let f^(-1)(x) be the Fibonacci number closest to x, and g^(-1)(s) be the number of the Fibonacci number s. Then, if we take g^(-1)(x)<n_0, the lowest number of prime numbers up to x can be expressed as follows.

4. lower bound on the number of primes when x>=n_0

If we take n such that n>n_0, then for each interval [f(n+k)^2, f(n+k+1)^2], (k: natural number), the minimum number of primes is multiplied by φ. In the interval after f(n_0), if the number of intervals is k, the lower bound on the overall number of primes can be expressed as an identity series up to k with the first term being 2-2φ^(-3)[*1] and the common ratio The lower bound on the number of primes in the whole can be expressed as an identity series up to k, where the first term is 2-2φ^(-3)[*1] and the common ratio is φ.

[In chapter 1, you saw that the covering of the interval [f(2m-1),f(2m+1)] consisting of Fibonacci numbers and the interval [f(n+1)^2, f(n+2)^2] consisting of the square numbers of Fibonacci numbers are almost the same.

If you take it as g^(-1)(x)>n_0, then the minimum number of prime numbers from n_0 to x is as the following.

5. singular functions

Singular functions are used, for example, in mechanics of materials. For details, please refer to reference [5].

In this article, we will use this singular function to calculate

Function α: returns 1 when x is less than n_0 and 0 when x is less than n_0

Function β: returns 0 when x is less than n_0, and 1 when x is greater than n_0

Create two functions.

Function α: 1-<x-n_0>

Function β: <x-n_0>.

By multiplying the expressions derived in Chapters 3 and 4 by the functions α and β, respectively, and adding them together, we get a function that calculates the lowest number of primes up to an arbitrary number x, i.e., the lower limit of the prime number calculation function π(x)[6].

6. summary (review of previous lesson)