New product formula for
Mohammad Bakkar
The National Center for Distinguished
September 22,2020
Abstract
This paper introduces a new product formula for the number using Euler
product and Wallis product. First, we will find a special case of the formula using
the result from the Basel's problem. Then we will test the validity of the formula on
computing program. Finally, we will find a generalization for this formula using the
zeta function for the even numbers. The derived formula connects with the odd
composite numbers.
Key words:
zeta function:
Bernoulli's numbers:
1. Introduction
has always been the most famous figure in the whole mathematics. Since
the first spot for it in computing the area of the circle, it has showed in places that
they are not related to the circle at any obvious way. From that day forth, it has been
discovered tons of relations that include this magical number from infinite series or
products, infinite fractions, incredible estimations, and integrations.
2. Theorems that will be used in this article
Theorem (1) (Euler product formula):
∑ ∏
Theorem(2)(Basel's problem):
Theorem (3)(Wallis' Identity):
Theorem(4):
3. Finding the formula for
By using Euler product at we get:
∏ ∏
Multiplying (4) with (3) side by side we get :
(∏ )( ∏ )
Now let’s simplify this product
Notice for we get
So all the prime numbers will vanish out and remain the composite numbers except for the
term
We can now expand (6) we get:
(∏ )( ∏ )
( )( )
( ) ( ) ( )
Noticing that the remaining terms have the form of where is a composite odd
integer except for the first term so the new formula is:
4. Testing the relation
The following program is done with C++ language. It's computing the product in (6) for large
numbers up to
#include <iostream>
#include <math.h>
#include <bits/stdc++.h>
using namespace std;
bool f(int x){
int t=0;
for(int i=1;2*i+1<x;i++){
int s=2*i+1;
if(x%s==0){
t++ ;
break;
return t;}
int main()
cout<<fixed<<setprecision(10);
long double l=1.0;
for(int i=4;i<100000;i++){
long double n=2*i+1;
if(f(n)==1){
long double W=((n-1)/n)*((n+1)/n);
l=l*W; }
cout<<l<<endl;
return 0;}
The output was: while the estimated value of
the two values are very close so this confirms the validity of the relation
5. Generalizing the product
Notation: we will call a number -composite if it's not written as where is a
prime number and is a positive integer.
Plugging in (1) to get:
Now, multiplying (8) with (3) side by side:
(∏ ) ( ∏ )
To simplify this product first notice that when we get
So every prime number to the power will vanish out. Now we can simplify the product at (9):
(∏ ) ( ∏ )
( )( )
( )( )
Noticing that the remaining terms have the form where is not an odd
prime raised to the power except for the first term so we can write the last product
as
Plugging this into (9) to derive the generalized formula:
6. Conclusion
We find that can be written as infinite product using non-prime numbers,
which is a new formula can be added many other formulas.
Also we find a generalization that connects , the even terms of Bernoulli's
numbers, and the -composite numbers.
7. References
1. Th. Rassias, Michael. Problem-Solving and Selected Topics in Number Theory.
Department of Pure Mathematics and Mathematical Statistics, University of
Cambridge, Cambridge CB3 0WB, New York, UK Springer, Dordrecht
Heidelberg London,2010, mthrassias@yahoo.com.
2. R. Dunbar, Steven. Topics in Probability Theory and Stochastic Processes.
Department of Mathematics 203 Avery Hall University of Nebraska- Lincoln, NE
68588-0130, November 2, 2017 http://www.math.unl.edu
3. Fine, Benjamin, and Gerhard Rosenberger. Number theory an introduction via
distribution of primes. Boston, Birkh¨auser, 2007
