# Questions tagged [riemann-hypothesis]

Questions on the Riemann hypothesis, a conjecture on the behavior of the complex zeros of the Riemann $\zeta$ function. You might want to add the tag [riemann-zeta] to your question as well.

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### Count the number of ordered triples of positive integers whose product is not greater than a given number?

Given N, count the number of ordered triplets(a,b,c) whose product $abc \leq N$. I have found the series here . But I am not sure I understand Benoit Cloitre work which proposed an efficent way to ...
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### Absolute convergence of a double matrix and of $\frac{1}{\zeta(\frac{1}{2}+\epsilon)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{\frac{1}{2}+\epsilon}}$

According to Wikipedia the convergence of the right hand side of: $$\frac{1}{\zeta(\frac{1}{2}+\epsilon)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{(\frac{1}{2}+\epsilon)}}$$ for $\epsilon$ an arbitrary ...
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### Density and distributions of those numerically or analytically KNOWN solutions of Riemann $\zeta(1/2 + r i)=0?$

We know the conjecture about the Riemann hypothesis is about the nontrivial zeros are on $$(1/2 + r i)$$ for some $r \in \mathbb{R}$ of Riemann zeta function. My question is how much is known about ...
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### The Riemann hypothesis as a statement about natural numbers

In this answer it is claimed that the Riemann hypothesis can be expressed as a statement about natural numbers. How would that look like?
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### Prove $\int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x}$ is Entire

Define $$f(s)= \int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x}$$ where $\psi(x)= \sum_{n=1}^{\infty}e^{-n^2\pi x}$ is the Jacobi Theta function. Claim- $f(s)$ is Entire My Try :- ...
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### Why is proving the Riemann Hypothesis so hard?

The Riemann Hypothesis is considered by many to be the most important unsolved problem in pure mathematics. Several attempts have been made in the last 150 years (here some of them are reported). RH ...
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### Is it hopeless to try and solve this equation analytically?

Can this equation be solved with analytical methods, or is it only numeric methods since current mathematical tools don't go that far? Its complex roots are the same as the roots of the zeta function ...
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### Direct derivation of $J(x) = \operatorname{li}(x) - \sum_{\rho} \operatorname{li}(x^\rho) + \int_{x}^{\infty} \frac{dt}{t(t^2-1)\log t} -\log 2$
I'm reading the Riemann's article on the number of prime numbers with a detailed explanation by W. Dittrich, and stucked to prove the Riemann's final formula  J(x) := \sum_{p^n \leq x} \dfrac{1}{n} =...