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I'm pretty sure the limitation value exists for the following expression. I have tried to factorize each item but to no avail. Any suggestions?

$(1-1/2)(1-1/4)(1-1/8)(1-1/16)$...

Golden_Ratio
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Curious George
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1 Answers1

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The Euler function is defined as

$$\phi(q)=\prod_{k=1}^\infty (1-q^k),\quad |q|<1.$$

More generally, the $q$-shifted factorial is defined as

$$(a;q)_\infty =\prod_{k=0}^\infty (1-aq^k),\quad |q|<1.$$

Your product is just $(1/2;1/2)_\infty=\phi(1/2)$.

Golden_Ratio
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  • I understand how to calculate \phi(n), but how is \phi(1/n) related to the former? – Curious George Mar 12 '22 at 06:16
  • @CuriousGeorge I don't understand your question. Please clarify. – Golden_Ratio Mar 12 '22 at 06:19
  • You used the totient of a fraction 1/2, but I only know how to calculate that for an integer in number theory, for example, T(5)=5(1-1/5). What about T(1/5), T(1/3). Any good links may help. Thanks. – Curious George Mar 12 '22 at 06:22
  • @CuriousGeorge It is NOT the totient function. Please see the two wiki links. – Golden_Ratio Mar 12 '22 at 06:22
  • I tried to make sense of the two wiki links, but it seems to be beyond my limited calculus knowledge, as I don't know how to use the fomular for different a/q values.. And I have not found an article that explains this from the basics. I thought it can be solved by some high school algebra tricks. – Curious George Mar 12 '22 at 07:01
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    @CuriousGeorge I don't think it has a known closed form. Some properties of the Euler function are known and special values of the Euler function have a nice closed form obtained by Ramanujan (as mentioned in the wiki link), but unfortunately, $\phi(1/2)$ is not among them. – Golden_Ratio Mar 12 '22 at 07:04
  • How about ϕ(1/3)? (1−1/3)(1−1/9)(1−1/27)(1−1/81)... – Curious George Mar 12 '22 at 07:08
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    @CuriousGeorge Again, you can get approximations but I am not aware of a closed form. This question was also asked here – Golden_Ratio Mar 12 '22 at 07:15