# Tagged Questions

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### Montgomery&Vaughan's Multiplicative number theory theorem 13.3

I can't understand well the proof of theorem 13.3 There exist a constant $C>0$ s.t. if RH is true, then for every $x\ge 2$ the interval $(x,x+Cx^{1/2}\log x)$ contains at least $x^{1/2}$ prime ...
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### Error term of the prime number theorem in arithmetic progressions

It is known that if $(a, q)$ and $q\le (\ln x)^N$, then the following is true $$\sum_{k\le x, k\equiv a\pmod{q}}\Lambda(k) = \frac{x}{\phi(q)} + O(x\exp(-C\sqrt{\ln x}))$$ where $C$ depends only on ...
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### Prime Counting: Relationship between Chebyshev's function and the Prime counting function

How do I show that if $\psi(x)=x+O(x^{1/2}\log^2(x))$ then $\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$ Where $\psi(x)$ is Chebyshev's second function and $\pi(x)$ is the prime counting ...
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### Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?

This is follow up question on this: How does $\sum_{p<x} p^{-s}$ grow asymptotically for $\mathrm{Re}(s) < 1$? There it is stated that:  \sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + ...
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### Where can I find the paper by Guy Robin?

$$\sigma(n) < e^\gamma n \log \log n$$ In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ...
### Nonnegativity of the quadratic Dirichlet L-function $L(\tfrac{1}{2},\chi)$ under GRH
I have been looking for a proof of the statement: "Assume the Generalized Riemann Hypothesis. Let $d$ be a fundamental discriminant and $\chi_d$ the associated primitive quadratic character. Then, ...