# Tagged Questions

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### Integration of complex function with respect to complex variable

I was given as homework to calculate the complex integral limit $$\lim_{T\rightarrow \infty} \frac {1}{2\pi i}\int_{c-iT}^{c+iT}\frac {x^s}{s^{k+1}}ds$$ where $c>0$ and $k\geq1$ is an integer. ...
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### Application of Dirichlet Theorem in AP to elementary number theory problems.

I have learnt this theorem in my class, however, "elementary" examples are very limited. (focusing more on analytic machinery) Are there any interesting applications to elementary number theory that ...
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### On Dirichlet Theorem on primes in AP.

Let $A(h,k) = \{h + km: m = 0,1,2,\dots\}\;\;$ (EDIT: and $(h,k)=1$) Without using Dirichlet's Theorem, Prove that for every positive integer $n$, $A(h,k)$ contains infinitely numbers relatively ...
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### Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
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### Trouble Identifying a “Psi” Function in Number Theory

In these lecture notes on number theory I am reading I came across the notation $\Psi(e^t;a,q)$ in connection with the Dirichlet theorem on arithmetic progression. I was hoping someone could help me ...
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### Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
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### How to prove that $\sum_{p \leq x} {\log p \over p} = \log x + O(1)$? [duplicate]

Problem Prove that $$\sum_{p \leq x} {\log p \over p} = \log x + O(1)$$ as $x \to \infty$. Notes: $p$ ranges over primes, $\log$ is natural Progress Using Riemann-Stieltjes integration and ...
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### How to prove this inequality $\pi(x) > \log x - 1$ involving the prime counting function?

Problem Prove that $\pi(x) > \log x - 1$. Progress Based on a hint and very elementary methods, I got that $$\prod_{p \leq x} (1-p^{-1})^{-1} \leq \prod_{k=2}^{\pi(x)+1} (1-k^{-1})^{-1}.$$ The ...
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### How to prove $\sum_p {1 \over p^s} = \sum_{n=1}^\infty {\mu(n) \over n} \log \zeta(ns)$?

Problem Prove that for $\operatorname{Re}(s)> 0$, $$\sum_p {1 \over p^s} = \sum_{n=1}^\infty {\mu(n) \over n} \log \zeta(ns),$$ where the sum extends over all primes $p$. Notes: $\log$ is ...
Problem Show that $$\gamma = \tfrac 12 \log 2 + {1 \over \log 2} \sum_{n=2}^\infty (-1)^n {\log n \over n}.$$ Progress I tried writing the terms $1/k$ of the harmonic sum in the definition of ...