# Why the zeta function?

Why is the zeta function, $\zeta(s)$ used to obtain information about the primes, namely giving explict formula for different prime counting functions, when there are many other functions that encode information about primes?

For example, the relation: $$\frac{-\zeta'(s)}{\zeta(s)}=\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$$

Could be seen as a special case of a polylogarithm identity: $$\frac{d}{ds}\text{Li}_s(x)=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^s}\text{Li}_s(x^n)$$

Where $x=1$.

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Are you going to finish the question? "Where $x=1$," --- what? –  Gerry Myerson Jan 27 '13 at 4:08
I was going to list other examples, but I don't think it really matters, so I deleted the coma –  Ethan Jan 27 '13 at 4:28
You might want to replace it with a period. It looks like you are in the middle of editing the question (and also it's typographically offensive). –  Potato Jan 27 '13 at 4:39

Ok so to ask you a question, I might ask: "Why don't we use category theory to study primes?".

Your answer would probably be something along the lines of "There aren't many known connections between these two objects that links them in a useful way".

This answers your question essentially, yes there are other mathematical objects connected with primes that tell us nice things about them but the Riemann zeta function has just turned out to be the best at telling us useful things so far.

Why would we want to study objects which don't appear to be connected with them (unless we had reason to believe that there should be a link)?

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The Euler product encodes unique factorization of natural numbers. A generalization of the zeta function exist for number fields and in that situation the Euler product there encodes the unique factorization of ideals.

Euler's debut was the solution of the Basel problem - he in fact evaluated the zeta function at even values. Euler proved the divergence of $\sum\frac{1}{p}$ using the Euler product in 1737, which was the first genuinely new proof that there are infinitely many primes in decades. He never (explicitly?) analytically continued the function.

Dirichlet proved that there are infinitely many primes in arithmetic progressions in 1837 as well as the class number formula using a version of the zeta function that has a periodic phase in the numerator, both of which require analytic continuation.

Riemann of course studied the function in much more depth, derived its functional equation and published On the Number of Primes Less Than a Given Magnitude in 1859 which insight into prime numbers and led to the proof of the prime number theorem.

A lot of difficult results in number theory have been strengthened on the assumption of the Riemann hypothesis, and there have been many many deep results proved using techniques related to zeta functions. So there is clearly something interesting going on there.

and http://www.dpmms.cam.ac.uk/~wtg10/zetafunction.ps tries to motivate it from "you could have discovered zeta" point of view.

• February 8: psi.pdf: Complex analysis enters the picture via the contour integral formula for \psi(x) and similar sums

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