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I took some notes on the historical use of zeta functions in number theory here Why the zeta function? but from looking up the dates I didn't realize something:

I remember reading that its not called the Euler zeta function because he never analytically continued it - even though Euler did lots with it, including studying primes.

But it seems that Dirichlet did, and must he not have also found the functional equation? So why did Riemann think to study the zeta function? wasn't he inspired by Dirichlet's work as well as Eulers?

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Riemann was indeed inspired by the work of Dirichlet. Googling "Riemann 1859" will bring a bunch of historical accounts, from the original manuscript to a Master's thesis on the history of the paper. // Lots of mathematical objects could justifiably be named after several mathematicians. It's not a race with a photo-finish. // You don't have to be the first to study an object in order to get it named after you. Sometimes "all you have to do" is to create something mind-blowing with this object, like an explicit formula for the prime-counting function. – user53153 Feb 2 '13 at 21:35
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Dirichlet characters and their L-series were introduced by Johann Peter Gustav Lejeune Dirichlet, in 1831, in order to prove Dirichlet's theorem on arithmetic progressions. He only studied them for real s and especially as s tends to 1. The extension of these functions to complex s in the whole complex plane was obtained by Bernhard Riemann in 1859.

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