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 $x=1$.
 A: This is a slightly misguided question. 
The zeta is used to find information about the primes because it gives information about the primes. There is no real reason a priori why one should look at it, and people have looked at every other function from which they could extract information about primes. Zeta is famous because people have been particularly successful at extracting information from it.
With time, a whole mass of related results have been obtained, and nowadays zeta functions appear everywhere in various forms, but this is not because of a desire to shun other information-encoding functions, but simply for the reason that there is an immense toolkit of methods, techniques, heuristics and so one that were developed in order to extract information from functions of that type.
A: 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.
Good read: https://terrytao.wordpress.com/2009/09/24/the-prime-number-theorem-in-arithmetic-progressions-and-dueling-conspiracies/
and http://www.dpmms.cam.ac.uk/~wtg10/zetafunction.ps tries to motivate it from "you could have discovered zeta" point of view.

http://www.math.harvard.edu/~elkies/M229.09/index.html 


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




A: 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)?
