In this document, I will show a new formula for Euler product formula, not equal to Riemann zeta function.

1. New formula for Euler product not equal to
Riemann z`eta function
Chris De Corte
chrisdecorte@yahoo.com
December 18, 2014
1 Introduction
In a previous writing [3], I already showed that, contrary to what is generally
accepted [1], the Riemann z`eta function does not equal the Euler product
or that:
p prime
1
1 − p−s
=
∞
n=1
1
ns
(1)
2 New formula
I am now glad to announce that the formula using the Euler product, without
using the complex power s, should be:
∀pi≤x
(1 − 1/pi) =
x
ln(x) · e
x
i=1(1/i)
(2)
1

2. 3 testing New formula
The accurateness of formula 2 can easily be shown by pasting the following
code in PARI/GP:
forstep(x=1000,41000,3000,print(prime(x),” ”,prime(x)/log(prime(x))/
exp(sum(i=1,prime(x),1/i,0.))/prodeuler(y=2,prime(x),(1-1/y))))
The result of this testing can be found in Figure 1.
4 Proof of formula 2
Formula 2 can be reproduced after eliminating γ from my definition of the
logarithmic function [4]:
ln(x = pi) =
1
eγ · x=pi
i=2 (1 − 1/pi)
(3)
using the definition of the Euler Mascheroni constant [2]:
γ = lim
n→∞
n
k=1
1
k
− ln(n) (4)
5 summary
We have found a new formula for the Euler product. This formula is gen-
erally better for higher x but is still acceptable for lower x. So, x, doesn’t
need to go to ∞ for the formula to be valid!
6 References
1. Wikipedia, the free encyclopedia (2014). Proof of the Euler product
formula for the Riemann zeta function. Page 1-2.
2. Wikipedia, the free encyclopedia (2014). Euler-Mascheroni constant.
Page 1.
2

3. Figure 1: This figure represents the fraction of the right side of formula 2 di-
vided by the left side of formula 2 for different primes (x). The accurateness
can be easily checked as 1 would be a perfect match
3

4. 3. Chris De Corte (2014). Disprove of equality between Riemann zeta
function and Euler product. Page 1-12.
4. Chris De Corte (2014). Probabilistic approach to prime counting.
Page 7.
4