The Zeta Function: Finding a general formula Part 1

I will be writing about the zeta function and finding a general formula for $s=2n$ .All you need is a bit of knowledge in Calculus and Trigonometry.

Now let's get started.Remember the Taylor series for $sin x$ ?

$\sin x=x- \frac{x^3}{3!}+\frac{x^5}{5!}-...$

Now let's divide both sides by x.

$\frac{\sin x}{x}=1- \frac{x^2}{3!}+\frac{x^4}{5!}-...$

The zeros of the sine function occur when $x=k\cdot\pi$ , where k is an integer. So $\frac{\sin x}{x}=(1-\frac{x}{\pi})(1+\frac{x}{\pi})(1-\frac{x}{2\pi})(1+\frac{x}{2\pi})...=(1-\frac{x^2}{\pi^2})(1-\frac{x^2}{2^2\pi^2})...$

Now let's substitute $x=\pi\cdot s$

Then $\frac{\sin \pi.s}{\pi.s}=(1-s^2)(1-\frac{s^2}{2^2})...=\displaystyle\prod_{k=1}^{\infty} (1-\frac{s^2}{k^2})$

Nobody likes to deal with infinite products, but if we take the logarithm of a product, it becomes a sum!

$\ln\frac{\sin \pi.s}{\pi.s}=\ln (\displaystyle\prod_{k=1}^{\infty} (1-\frac{s^2}{k^2})) =\sum_{k=1}^{\infty} \ln(1-\frac{s^2}{k^2})$

That makes

$\ln\sin \pi.s=\ln (\pi.s)+\displaystyle\sum_{k=1}^{\infty} \ln(1-\frac{s^2}{k^2})$

Now if we differentiate both sides with respect to s (I will not go through the derivation of this, just bash with the chain rule a bit), we will get

$\pi.\frac{\cos \pi.s}{\sin \pi.s}=\frac{1}{s}+\displaystyle\sum_{k=1}^{\infty} \frac{1}{(1-\frac{s^2}{k^2})}.-\frac{2s}{k^2}$

Now we multiply both sides by $s$ and we get

$\pi.s\frac{\cos \pi.s}{\sin \pi.s}=1+\displaystyle\sum_{k=1}^{\infty} \frac{1}{(1-\frac{s^2}{k^2})}.-\frac{2s^2}{k^2}$ .

To be continued in Part 2.

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Comments

Really interesting. Now how can we necessarily states that $\frac{\sin x}{x}=\left(1-\frac{x}\pi\right)\left(1+\frac{x}\pi\right)\left(1-\frac{x}{2\pi}\right)\left(1+\frac{x}{2\pi}\right)\dots$ ?

Cody Johnson - 7 years, 1 month ago

Well, I cannot exactly prove it.It's like a continuation of the Fundamental Theorem of Algebra, but for infinite polynomials.

Bogdan Simeonov - 7 years, 1 month ago

In the line before you talk about differentiation, shouldnt you take the natural log of the product instead of making it a product of natural logs?

Isaac Thomas - 6 years, 2 months ago

Oh, sorry about that.Also it's weird that people still check out this post, it's like a year old :D

Bogdan Simeonov - 6 years, 2 months ago

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Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$