Expressing zeta(3) in terms of a Glaisher-Kinkelin-like constant

In summary, this conversation is about expressing $\zeta(3)$ in terms of a constant similar to the Glaisher-Kinkelin constant using the Euler-Maclaurin summation formula and the functional equation of the Riemann zeta function. The constant $B$ exists and can be found in several papers, and it is also possible to express $\zeta(5)$ in terms of a similar constant using a different representation of the Riemann zeta function.
  • #1
polygamma
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In a previous thread I showed how to express $\zeta'(-1)$ in terms of the Glaisher-Kinkelin constant.

http://mathhelpboards.com/challenge-questions-puzzles-28/euler-maclaurin-summation-formula-riemann-zeta-function-7702.html

This thread is about expressing $\zeta(3)$ (sometimes referred to as Apery's constant) in terms of a constant similar to the Glaisher-Kinkelin constant.

Specifically, $$\zeta(3) = 4 \pi^{2} \log B$$ where $$\log B = \lim_{n \to \infty} \left[ \sum_{k=1}^{n} k^{2} \log k - \left(\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6} \right) \log n + \frac{n^{3}}{9} - \frac{n}{12} \right] $$
Use the Euler-Maclaurin summation formula (or perhaps summation by parts) to show that the constant $B$ exists.Then using the representation of the Riemann zeta function derived in the other thread,

$$ \zeta(s) = \lim_{n \to \infty} \left( \sum_{k=1}^{n} k^{-s} - \frac{n^{1-s}}{1-s} - \frac{n^{-s}}{2} + \frac{s n^{-s-1}}{12} \right) \ \ \big(\text{Re}(s) > -3 \big) $$

show that

$$ \zeta'(-2) = - \log B $$Finally use the functional equation of the Riemann zeta function to show that $$ \zeta(3) = 4 \pi^{2} \log B $$
 
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  • #2
Let $f(x) = x^{2} \ln x$.

Then

$$ \sum_{k=1}^{n-1} f(k) = \sum_{k=1}^{n} f(k) - n^{2} \ln n = \int_{1}^{n} f(x) \ dx + B_{1} \Big(f(n) -f(1) \Big) + \frac{B_{2}}{2!} \Big( f'(n) - f'(1) \Big) $$

$$ + \frac{1}{3!} \int_{1}^{n} B_{3} (x - \lfloor x \rfloor) f^{'''}(x) \ dx$$$$ = \frac{x^{3} \log x}{3} - \frac{x^{3}}{9} \Big|_{1}^{n} - \frac{1}{2} \Big(n^{2} \ln n -0 \Big) + \frac{1}{12} \Big(2n \ln n + n -1 \Big) + \frac{1}{6} \int_{1}^{n} B_{3} (x - \lfloor x \rfloor) \frac{2}{x} \ dx$$

$$ = \frac{n^{3} \log n}{3} - \frac{n^{3}}{9} + \frac{1}{9} - \frac{n^{2} \ln n}{2} + \frac{n \log n}{6} + \frac{n}{12}- \frac{1}{12} + \frac{1}{3} \int_{1}^{n} \frac{B_{3} (x - \lfloor x \rfloor)}{x} \ dx$$$$ \implies \sum_{k=1}^{n} k^{2} \ln k - \Big( \frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6} \Big) \ln n + \frac{n^{3}}{9} - \frac{n}{12} = \frac{1}{36} + \frac{1}{3} \int_{1}^{n} \frac{B_{3} (x - \lfloor x \rfloor)}{x} \ dx$$Now take the limit of both sides of the equation.

The integral $ \displaystyle \int_{1}^{\infty} \frac{B_{3} (x - \lfloor x \rfloor)}{x} \ dx$ converges (condtionally) by Dirichlet's convergence test for integrals.

Therefore,

$$ \log B = \lim_{n \to \infty} \left[ \sum_{k=1}^{n} k^{2} \log k - \left(\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6} \right) \log n + \frac{n^{3}}{9} - \frac{n}{12} \right] $$

exists.From the other thread,$$ \zeta'(s) = \lim_{n \to \infty} \Bigg[- \sum_{k=1}^{n} k^{-s} \log k - \frac{-n^{1-s} (1-s) \log n +n^{1-s}}{(1-s)^{2}} + \frac{n^{-s} \log n}{2} $$

$$ + \frac{1}{12} \left(n^{-s-1}- sn^{-s-1} \log n \right) \Bigg] \ \ (\text{Re}(s) > -3)$$Plug in $s=-2$ to get

$$ \zeta'(-2) = \lim_{n \to \infty} \Bigg[- \sum_{k=1}^{n} k^{2} \log k - \frac{-3n^{3} \log n +n^{3}}{9} + \frac{n^{2} \log n}{2} + \frac{1}{12} \left(n+2n \log n \right) \Bigg] $$

$$ = \lim_{n \to \infty} \left[ -\sum_{k=1}^{n} k^{2} \log k + \left(\frac{n^{3}}{3} + \frac{n^{2}}{2} + \frac{n}{6} \right) \log n - \frac{n^{3}}{9} + \frac{n}{12} \right] = - \log B$$Next differentiate the functional equation of the Riemann zeta function.

$$ \zeta'(s) = \log (2) 2^{s} \pi^{s-1} \sin \left( \frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)+ \log(\pi) 2^{s} \pi^{s-1} \sin \left( \frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)$$

$$ + \frac{\pi}{2} 2^{s} \pi^{s-1} \cos \left( \frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s) - 2^{s} \pi^{s-1} \sin \left( \frac{\pi s}{2} \right) \Gamma'(1-s) \zeta(1-s)$$

$$ -2^{s} \pi^{s-1} \sin \left( \frac{\pi s}{2} \right) \Gamma(1-s) \zeta'(1-s)$$At $s=-2$,

$$ \zeta'(-2) = \frac{\pi}{2} 2^{-2} \pi^{-3} \cos \left( -\pi \right) \Gamma(3) \zeta(3) = -\frac{\zeta(3)}{4 \pi^{2}}$$Which implies

$$ \zeta(3) = -4 \pi^{2}\zeta(-2) = 4 \pi^{2} \log B $$
As far as I know, the constant $B$ doesn't have a name. But it can be found in several papers.

I'm pretty sure you could express $\zeta(5)$ in terms of a similar constant. But that would require using a different representation of the Riemann zeta function.
 

FAQ: Expressing zeta(3) in terms of a Glaisher-Kinkelin-like constant

1. What is zeta(3)?

Zeta(3) is a mathematical constant that is also known as Apéry's constant or the Apéry's constant. It is derived from the Riemann zeta function and has a numerical value of approximately 1.20206.

2. What is a Glaisher-Kinkelin-like constant?

A Glaisher-Kinkelin-like constant is a mathematical constant that is closely related to the Glaisher-Kinkelin constant, which is a special value of the Barnes G-function. These constants are used in number theory and have connections to the Riemann zeta function.

3. How is zeta(3) expressed in terms of a Glaisher-Kinkelin-like constant?

Zeta(3) can be expressed in terms of a Glaisher-Kinkelin-like constant as zeta(3) = zeta(3) * e^(C), where C is the Glaisher-Kinkelin-like constant. This means that zeta(3) can be written as a product of itself and the exponential of the Glaisher-Kinkelin-like constant.

4. Why is expressing zeta(3) in terms of a Glaisher-Kinkelin-like constant important?

Expressing zeta(3) in terms of a Glaisher-Kinkelin-like constant is important because it allows for a more compact and elegant representation of the constant. It also establishes a connection between zeta(3) and other important mathematical constants, which can help in further mathematical research and explorations.

5. Are there any practical applications of expressing zeta(3) in terms of a Glaisher-Kinkelin-like constant?

While the expression of zeta(3) in terms of a Glaisher-Kinkelin-like constant may not have direct practical applications, it has significance in theoretical mathematics and can help in understanding the connections between different mathematical constants and functions. It can also lead to new insights and discoveries in number theory and other related fields.

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