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camilus
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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?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
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.
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.
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.
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.
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.