# Questions tagged [chebyshev-function]

For questions about Chebyshev functions $\vartheta(x)$ and $\psi(x)$, which are often used in number theory. For questions about Chebyshev polynomials, use the (chebyshev-polynomials) tag.

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### Establish an upper bound for $\zeta'(s)\over\zeta(s)$?

Using Perron's formula, I am able to show that $$\psi(x)={1\over2\pi i}\int_{a-i\infty}^{a+i\infty}\left[-{\zeta'(s)\over\zeta(s)}\right]{x^s\over s}\mathrm ds$$ where $\psi(x)$ is the Chebyshev's ...
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### How can we prove $\sum_{p\leq x}\ln(p) < 2x$?

I've been reading this paper for Merten's second theorem. Every thing is just fine but I think that theorem 11 (3.5.2) can have an easier proof. Is there any better approach or an explanation that why ...
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### $\sum_{p,m\geq 3}(-1)^{m(p-1)/2}e^{-p^my}\log p = O(y^{-1/3})$

Show that for sufficiently small $y$ we have $\sum_{p,m\geq 3}(-1)^{m(p-1)/2}e^{-p^my}\log p = O(y^{-1/3})$ where $m\geq 3$ represents all positive integers from $3$ onwards, while $p\geq 3$ ...
43 views

### How to show convergence of the series sum over zeroes of the zeta function

In the explicit formula for the Chebyshev function, one of the terms that appears is $$\sum_{\rho} \frac{x^\rho}{\rho}$$ where the sum is taken over the nontrivial zeroes of the zeta function. I'm a ...
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### Big O and Asymptotic Notation [closed]

So I am currently working in the Apostol Book "Introduction to Analytic Number Theory", and am attempting to prove one of the major results regarding the prime number theorem. Specifically I am trying ...
181 views

### Proof of increasing positive valued function

Let $g$ be a monotonically increasing positive valued function defined on $\mathbb{R}$. Show that $\mathbb{P}(X\geq a) \leq \frac{\mathbb{E}[g(X)]}{g(a)}$. -I've tried expanding $\mathbb{E}[g(x)]$ ...