# Tag Info

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### Proof of Dirichlet's Theorem on Primes using $\sum_{\substack{p=1\\ p\equiv h\bmod k}}^\infty\frac{\ln p}{p^s}\sim\frac{1}{\varphi(k)(s-1)}$

The major aspects of the proof are implicit in most treatments. I don't know where this is written explicitly, but I'll sketch a proof of the claim and how this implies Dirichlet's Theorem. The main ...
Accepted

### Can $f(z) = \sum_p \frac {z^p}{p^2}$ teach us about primes?

Today the prime number theorem is proved using the Dirichlet series $\sum \Lambda(n)/n^s$, where $\Lambda$ is the von Mangoldt function. This is the negative log derivative $-\zeta'(s)/\zeta(s)$, ...

### Complex logarithm of zeta functions and L-functions

You ask a good question that is often misunderstood and not explained clearly in textbooks. In brief, the left side of your log equation is defined to be the right side and there is no need to refer ...
Accepted

This is a good, natural question. The point is that the measure $\mu(z) = \frac{dx dy}{y^2}$ is the unique (up to multiple) Haar measure here. This measure is invariant under the isometries of the ...
1 vote

1 vote

### The inverse of a Dirichlet product is the Dirichlet product of the inverses of each function

Actually we can look at this question from the group theory perspective. If $a,b$ are elements of an abelian group $G$, then $(a*b)^{-1}$ is $a^{-1} * b^{-1}$. So we just need to prove that all ...

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