# Questions tagged [riemann-hypothesis]

Questions about the famous conjecture from Riemann saying that the non-trivial zeroes of the Riemann Zeta function all lie on the so-called critical line $\Re(s)=\dfrac{1}{2}$, its various generalizations and the different approaches towards its solution.

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### Prime differences and zero multiplicity

Concerning gaps between consecutive primes, Paul Erdős conjectured that: $$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$ Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
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### Tools to prove lower bounds in analytic number theory

Perron's formula and related methods are used to relate statements such as the Riemann hypothesis to upper bounds of functions occurring in analytic number theory. For example, Perron's formula is ...
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### What is the proof for any non trivial zero? [closed]

There are many known nontrivial zeros of the Riemann Zeta function, but I have never seen proof that any of them actually resolve to zero. The trivial zeros make sense because there is a more ...
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### GRH and the Euler product

Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
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### Contribution of Yitang Zhang latest results if correct to correlation conjecture of H. L. Montgomery?

There are some integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. One of them is the integral introduced by Selberg related to estimating the variance of primes in ...
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### An approach to the prime number theorem with Rademacher variables and a recursive formula for the prime pi function?

Consider the bipartite graphs defined here: Why is this bipartite graph a partial cube, if it is? We do random walks on them with equal propability and since the graphs are finite and connected the ...
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### Robin's inequality for odd numbers

In this article (Theorem 1.2) there is a proof for Robin's inequality for odd numbers, $\sigma(n)/n< e^{\gamma}\log(\log(n))$ where $\gamma$ is the Euler-Mascheroni constant and $\sigma(n)$ is the ...
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### Example of sequence of graphs which satisfy the Riemann hypothesis or the prime number theorem?

Let us look at the sequence of bipartite graphs $G_n = (V_n, E_n)$ where $V_n = A_n \cup B_n$ defined in this quesiton: Why is this bipartite graph a partial cube, if it is? . The shortest path ...
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### Formalisation of the Riemann Hypothesis

Might there be a research team that has formalised the Riemann Hypothesis? So far I have encountered two related questions: Is there a formulation of the Riemann Hypothesis in first-order arithmetic? ...
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### Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?

The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the ...
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### On sum over non trivial zeroes of riemann zeta function

I wanted to know if there is an estimate or any closed form on the following partial sum series $$\sum_{n=1}^k \frac{1}{|\alpha_n||(\zeta'(\alpha_n))|}$$ Where "$\alpha_n$" runs over all non-...
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### Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false? One interesting example, and the impetus for this question, is work in number ...
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### Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry

Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
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### Colossally abundant numbers and the Riemann hypothesis

[This question followed up from a question on Math StackExchange.] Writing Robin's inequality for the Riemann hypothesis (RH) as $$\frac{\sigma(n)}{n \ln\ln n} < e^\gamma \;,$$ we can take ...
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A weaker version of the Riemann hypothesis is the claim that if $\zeta(s) = 0$ then $Re(s) \leq 1 - h$ for some constant $h> 0$. What would the consequences be of a result of this type?