What is the connection between Euler's formula and the Zeta function?

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In summary: For example, the number 3 is the product of 2 and 5, so the sum of all 3's is 12. The number 5 is the product of 2 and 3, so the sum of all 5's is 10. The sign of a product is determined by the order in which the primes are multiplied. For example, the number 3 is the product of 2 and 5, so the sign of 3 is positive because the order of the multiplication is positive. The number 5 is the product of 2 and 3, so the sign of 5 is negative because the order of the multiplication is negative.
  • #1
camilus
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Hi everyone. I'm trying to understand the step where they wrote

1/2 ∏1/(1+p^-3) =1/2 Ʃ(-1)^ord(k)/k^3

How can I see this? I know the Euler product formula, but it has a negative sign before the p^-3, where here we have a + sign.

Thanks for the help.
 

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  • #2
The step going from the second to last line, to the last line.
 
  • #3
I need help understanding this equality:

[itex]\prod_{p-prime} \frac{1}{1+\frac{1}{p^3}}= \sum_{k=1}^\infty \frac{(-1)^{\sum_p ord_p(k)}}{k^3}[/itex]


Any help is greatly appreciated!
 
  • #4
We know
$$\prod_\mathbb{P} (1-p^{-s})^{-1}=\prod_\mathbb{P} \left( \sum_{\mathbb{Z} \ge 0} p^{-s n} \right) ^{-1} = \sum_{\mathbb{Z}>0} n^{-s}$$

with the minus sign it is a little more complicated

$$\prod_\mathbb{P} (1+p^{-s})^{-1}=\prod_\mathbb{P} \left( \sum_{\mathbb{Z} \ge 0} (-p)^{-s n} \right) ^{-1} = \sum_{\mathbb{Z}>0} (-1)^{\sum ord_p(k)}n^{-s}$$

Where the ord_p(k) makes sure we get the right sign (it counts the minuses), clearly

$$\left( \prod_\mathbb{P} (1-p^{-s})^{-1} \right) \left( \prod_\mathbb{P} (1+p^{-s})^{-1} \right) =\zeta(2n)$$

but instead of using that your link makes a simple estimate
 
  • #5
lurflurf said:
We know
$$\prod_\mathbb{P} (1-p^{-s})^{-1}=\prod_\mathbb{P} \left( \sum_{\mathbb{Z} \ge 0} p^{-s n} \right) ^{-1} = \sum_{\mathbb{Z}>0} n^{-s}$$

with the minus sign it is a little more complicated

$$\prod_\mathbb{P} (1+p^{-s})^{-1}=\prod_\mathbb{P} \left( \sum_{\mathbb{Z} \ge 0} (-p)^{-s n} \right) ^{-1} = \sum_{\mathbb{Z}>0} (-1)^{\sum ord_p(k)}n^{-s}$$

Where the ord_p(k) makes sure we get the right sign (it counts the minuses), clearly

$$\left( \prod_\mathbb{P} (1-p^{-s})^{-1} \right) \left( \prod_\mathbb{P} (1+p^{-s})^{-1} \right) =\zeta(2n)$$

but instead of using that your link makes a simple estimate
Should [itex]\prod_\mathbb{P} \left( \sum_{\mathbb{Z} \ge 0} p^{-s n} \right) ^{-1}[/itex] have that ^(-1) after it? Or am I missing something..? Are you rewriting 1/(1-p^-s) using geometric series?

Anyways, thanks that was very helpful, I'm looking into the proofs of the product formula via this route.
 
  • #6
Yes that is a geometric series expansion.
 
  • #7
What about the negative one?
 
  • #8
$$\prod_\mathbb{P} (1+p^{-s})^{-1}=
\prod_\mathbb{P} (1-(-p^{-s}))^{-1}= \\
\prod_\mathbb{P} \left( 1-p^{-s}+p^{-2s}-p^{-3s}+...+(-1)^k p^{-ks}+... \right) ^{-1} = \sum_{\mathbb{Z}>0} a_n n^{-s}=\frac{\zeta(2n)}{\zeta(n)}$$
a_n being in{-1,1}
In the sum each term is positive or negative and we can determine which using
ord_p(k)
it is a bit tedious to compute, which is why the link estimates
here is the wikipedia
[PLAIN]http://en.wikipedia.org/wiki... one way (fundamental theorem of arithmetic).
 
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1. What is Zeta(3) and Euler's formula?

Zeta(3) is a mathematical constant that is also known as the Apéry's constant. It has a value of approximately 1.2020569 and is closely related to the Riemann zeta function. Euler's formula, also known as Euler's identity, is a mathematical equation that relates the five most important mathematical constants: 0, 1, π, e, and i.

2. What is the significance of Zeta(3) and Euler's formula?

Zeta(3) and Euler's formula have several important applications in mathematics and physics. For example, Zeta(3) appears in various areas of mathematics such as number theory, algebraic geometry, and algebraic topology. Euler's formula is used in complex analysis, differential equations, and signal processing.

3. How is Zeta(3) calculated?

Zeta(3) can be calculated using different methods such as the Euler-Maclaurin formula, the Riemann-Siegel formula, and the Levinson-Lessman formula. These methods involve complex mathematical equations and algorithms, and the calculation of Zeta(3) has been an area of research for many mathematicians.

4. What is the relationship between Zeta(3) and the Riemann zeta function?

The Riemann zeta function is a more general function that includes Zeta(3) as a special case. Zeta(3) is the value of the Riemann zeta function when the input is 3. This function has many important properties and has been extensively studied by mathematicians, particularly in relation to the distribution of prime numbers.

5. Can Zeta(3) and Euler's formula be used to solve real-world problems?

Yes, Zeta(3) and Euler's formula have practical applications in various fields such as engineering, physics, and computer science. For example, Euler's formula is used in signal processing to convert periodic signals into complex numbers, which can then be manipulated using mathematical operations. Zeta(3) has also been used in the development of efficient algorithms for data compression and cryptographic systems.

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