In summary, the goal of this conversation was to get closer to the values of the zeta function (ζ(s)) and eta function (η(s)) for odd values of s, through expanding on previous insights and using Fourier series. The author's patience is appreciated as there are many equations to load on the page and they specifically mention calculating sums involving fractions with odd powers.
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Svein
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The goal is to get a little bit closer to the values of the zeta function (ζ(s)) and the eta function (η(s)) for some odd values of s. This insight is an expansion of two of my previous insights (Further Sums Found Through Fourier Series, Using the Fourier Series To Find Some Interesting Sums). You patience is appreciated as there are many equations to load on this page.
Specifically I will calculate the sums

[itex]\frac{1}{1^{p}}-\frac{1}{3^{p}} +\frac{1}{5^{p}}-\frac{1}{7^{p}}… [/itex]
[itex]\frac{1}{1^{p}}+\frac{1}{2^{p}}-\frac{1}{4^{p}} -\frac{1}{5^{p}}+\frac{1}{7^{p}}…[/itex]
[itex]\frac{1}{1^{p}}+\frac{1}{2^{p}}+\frac{1}{3^{p}}...

Continue reading...
 
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Geoffrey Campbell
I think that Euler had already found these results by utilizing the infinite product for (sin x)/x and it's variations starting from an heuristic argument about the roots of an infinite polynomial (the power series in fact). The approach taken in this article is more modern and uses the fact of contemporary calculations of the coefficients, which is nowadays easy. A nice slant on this, however, unfortunately not giving us the desired zeta values for the odd numbers 3, 5, 7, etc.
 
  • #3
In the line before Section 5 you mention the eta function, presumably a misprint...
 
  • #4
A. Neumaier said:
In the line before Section 5 you mention the eta function, presumably a misprint...
No, it is not a misprint. The eta function resembles the zeta function, the difference lies in the sign of the even powers of 1/n (https://en.wikipedia.org/wiki/Dirichlet_eta_function).
 

1. What is the concept of "odd powers of 1/n" in mathematics?

The concept of odd powers of 1/n refers to the sum of all the odd powers of the reciprocal of a given number n. This means that we are adding all the values obtained by raising 1/n to odd powers, such as 1/n, 1/n^3, 1/n^5, and so on.

2. Why is it important to explore the sums of odd powers of 1/n?

Exploring the sums of odd powers of 1/n can help us understand the behavior of these sums and their relationship to the original number n. It also has applications in various areas of mathematics, such as number theory and calculus.

3. How can the sums of odd powers of 1/n be calculated?

The sums of odd powers of 1/n can be calculated using a formula, which involves the Bernoulli numbers and the harmonic numbers. This formula is known as the Euler-Maclaurin formula and it provides an efficient way to calculate these sums.

4. What are some real-life examples of the sums of odd powers of 1/n?

One example of the sums of odd powers of 1/n can be found in the study of electrical circuits, where it is used to calculate the total resistance of a circuit. It is also used in physics to calculate the total energy of a system.

5. What are some interesting patterns that can be observed in the sums of odd powers of 1/n?

One interesting pattern is that the sums of odd powers of 1/n tend to approach a specific value as n increases. This value is known as the Riemann zeta function and it has connections to prime numbers and the distribution of prime numbers.

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