Approximation of the constant c to make the equation relating prime numbers to pi and Napier numbers

0 References

[1] https://youtu.be/gKhR5yeiCqw "6. Pi Day Commemoration"

[2]http://cosmos.art.coocan.jp/cf/cf24.htm "Example of Expansion of a Number into a Series of Fractions"

[3]https://arxiv.org/pdf/math/0506319.pdf "DOUBLE INTEGRALS AND INFINITE PRODUCTS FOR SOME CLASSICAL CONSTANTS VIA ANALYTIC CONTINUATIONS OF LERCH'S TRANSCENDENT" JESU ́S GUILLERA AND JONATHAN SONDOW

1 Introduction to the video

In a previously uploaded video [1], it is introduced that there would be the following relationship between pi and prime numbers.

2 Numerical Results

The numerical experiment showed that the constant c would be approximately 2.00390625. The relative error was also found to be 0.00799435%.

3 Considerations from William Braunecker's Formula

We can also use William Braunecker's formula [2] to expand π/4 into a series of fractions, so we can express the Napier number using the constant c, the prime number p, and the natural numbers as follows

But how in the world can we determine the constant c?

4 Guillera-Sondow's Napier Number Function Display

In addition, there is the following Guillera-Sondow display of the Napier number function [3].

Therefore, the following equation holds, since equality holds for infinite products of n.

It should be noted that for c, the equality is only valid for infinite products of n, and it is an approximation for finite products.

where p_n means the nth prime number.

5 Considerations on the properties of c

Clearly, c is an irrational number. However, it is not clear whether it is a transcendental number.