About an identity by Euler

In summary, Euler proved that the infinite product 1/1-p**(-s) with p running over all primes was equal to R(s) where R(s) is the Riemann z function. The question was raised if the product 1/1-exp(-sp) over all primes would be the same as summing the series 1+exp(-s)+exp(-2s)+..=1/1+exp(-s) or what would be the equality. The response was that this is almost certainly not the case, as the primes are multiplied together in the first product and added together in the second, making it unlikely for them to have the same identity. It was also mentioned that there is no known identity for
  • #1
eljose79
1,518
1
Euler proved that the infinite product

1/1-p**(-s) with p running over all primes was equal to R(s) where R(s) is teh Riemman z function my question is if the product:

1/1-exp(-sp) over all primes would be hte same as summing the series

1+exp(-s)+exp(-2s)+..=1/1+exp(-s) or what would be the equality...thanks.
 
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  • #2
Almost certainly not.

In the orginal case, imagine multiplying out some finite portion and seeing what happens.

they key thing to note is that the primes and up multiplied together, and you can sort of see how you get all the integers out of it.

in the second the primes just get added together because they are in the exponents, so expecting to get the identity you conjecture is extremely unlikey.
 
  • #3
and do you know where i could find the answer to know the product:

1/1-exp(-sp) porduct over all primes ?..
 
  • #4
no, I don't know any identity it satisfies. why on Earth should it even have a nicer form than that? have you even bothered to multiply out the first few terms to see what it looks like?
 

What is the concept of identity according to Euler?

Euler's concept of identity refers to the idea that an equation or mathematical statement remains true regardless of the values of the variables involved. In other words, the equation holds true for any value of the variables that satisfy the equation.

What is the significance of Euler's identity in mathematics?

Euler's identity, also known as the Euler's equation, is considered to be one of the most beautiful and profound equations in mathematics. It relates five fundamental mathematical constants - the number 0, 1, π, e, and i - in a single equation, highlighting the interconnectedness of these seemingly unrelated constants.

How did Euler come up with his concept of identity?

Euler's concept of identity can be traced back to his extensive work in algebra and calculus. He realized that certain equations and mathematical statements remained true regardless of the specific values of the variables involved, leading him to develop the concept of identity.

What are some real-world applications of Euler's identity?

Euler's identity has various applications in different fields of science and engineering. It is used in signal processing, electrical engineering, and quantum mechanics, to name a few. It also has practical applications in the design of electronic circuits and communication systems.

How has Euler's identity influenced the development of modern mathematics?

Euler's identity has had a significant impact on the development of modern mathematics. It has inspired further research and discoveries in fields such as complex analysis, number theory, and topology. It has also influenced the study of symmetry and patterns in mathematics.

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