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.

292 questions
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Proof by contradiction and ZF set theory

The following is quoted from Sir Michael Atiyah's draft proof of Riemann Hypothesis, Section 5, stating that 'To be explicit, the proof of RH in this paper is by contradiction and this is not accepted ...
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Theorem 14.25(A) in Titchmarsh “The theory of the Riemann zeta-function”

In Titchmarsh's book "The theory of the Riemann zeta-function" there's theorem 14.25(A) on page 369 of the second edition where a summand $1/\zeta(s)$ appears out of the blue, so it seems... Oh, I do ...
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What is the mass center of the Riemann Zeta Function across the critical line?

I just came with the idea: what is the center of mass of the Riemann Zeta Function across the critical line? I mean: when you plot the parametric graph across the critical line, you get the famous ...
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Is there an analytic function with zeroes only at $-2n$, and zeroes at $\frac12\pm it$, and further, symmetric zeroes within the critical strip?

Is there an analytic function with zeroes only at: every $-2n$, $\frac12\pm it$, and at least one at $\frac12\pm\epsilon\pm it$ where $0<\epsilon<\frac12, t\neq0$ (and these zeroes observing ...
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Unclear multiplication with Riemann zeta functional equation with eta

In the region 0 < Re(s) < 1 we know that $$\zeta(s) = 1/(1-2^{1-s}) \sum_1^\infty (-1)^{n+1}/n^s\,.$$ This is a multiplication of two complex numbers. Question 1: Am I right to suppose ...
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Clarrification Regarding the Robin Inequality

I just read a paper related to the Robin Inequality, and the abstract read: "Abstract. Let σ(n) denote the sum of divisors function. In this paper we give a simple proof of the Robin inequality (R): ...
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Questions and concerns

I would like to know if solving the riemann hypothesis as well as the twin prime conjecture are still questions within mathematics? I have been unable to find credible resources.
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Riemann-Zeta Zeros and Quasicrystals

I came across quasicrystals in the Wikipedia page for the Riemann Hypothesis and then followed the references. On page 215 of Birds and Frogs Dyson makes the claim If the Riemann hypothesis is true,...
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How is the Riemann zeta function $\zeta(s)$ determined by its values in a small open disc?

This question pertains to the following quote from chapter 7 section 9 of "Summing it Up: From One Plus One to Modern Number Theory" by Avner Ash and Robert Gross: "What makes the Riemann ...
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Is there a known increasing sequence of positive integers $\{\textbf{a}\} = a_0<a_1<a_2<.....$ such that all the zeros $z_k$ on $\Re[z]>0$ of the complex function $F(z;\{\textbf{a}\})= \... 0answers 43 views Permutation of natural numbers relative to the Mobius Function We can define a function$f:\mathbb{N}\rightarrow \mathbb{N}$such that:$f$is bijective. If$f(n)=m$, then$f(m)=n$. If$f(n)=m$then$\mu(n)+\mu(m)=0$We want to define$f$such that the ... 1answer 550 views Using linear algebra to study number theory? I've posted a paper on arXiv that outlines a linear algebra approach to number theory. Specifically, I have the following questions: Is it possible to draw connections between the factorization ... 0answers 100 views is$\zeta(s) = \frac{1}{1-2^{1-s}}\eta(s)$an analytic continuation of$\zeta(s)$for$\sigma > 0$It seems from what I have read on the net, that the above representation of$\zeta(s)$is a valid analytic continuation of$\sum_{i=1}^{\infty}\frac{1}{i^s}$for$\sigma > 0$except for a simple ... 1answer 233 views Riemann hypothesis and Robin's Inequality Implications Is the following statement true: Let$\Bbb{A}$be the set of all Natural numbers n, greater than or equal to 5041, for which the inequality$\displaystyle \sigma(n)<e^{\gamma}n\log\log n$is not ... 1answer 57 views What is meant by the term “discrete” number with respect to the imaginary part of the non-trivial zeros of$\zeta(s)$? The following link indicates the imaginary parts of the non-trivial zeros of the Riemann zeta function$\zeta(s)$are "discrete" numbers. New Insight into Proving the Riemann Hypothesis What is ... 1answer 238 views Riemann Hypothesis numeric verification question? As I found in wikipedia Riemann Hypothesys has been verified numerically by X. Gourdon (2004) up to 10000000000000 ($10^{13}$) zeroes. I have a few question about how they did it. I tried to read on ... 2answers 181 views An explanation of the importance of analytic formulas representating arithmetic functions related to equivalences to the Riemann Hypothesis I'm curios about the following question, from an informational viewpoint. What is the purpose in finding/getting analytic formulas for specific arithmetic functions in the context of the Riemann ... 1answer 86 views question on Riemann$\zeta(s)$I have a question that is troubling me. From the functional equation of$\zeta(s)$, can we not conclude that both$\zeta(s)$and$\zeta(1-s)$have the same non-trivial zeros (differing at most in ... 0answers 248 views Is there a hidden connection between RH and the golden ratio? I realized today that, considering the circle$ \Gamma_{\Delta} $on the Riemann sphere whose image through the stereographic projection is the critical line$ \Delta $, the affixes of the images of ... 0answers 79 views Does statement (1) imply statement (2)? [closed] Statement 1 : (Robin) proved that if the R.H. is false then there exist constants$0<\beta <\frac{1}{2}$and$c>0$small , such that$\sum \limits_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c ...
The problem statement, all variables and given/known data Question Use the functional equation to show that for : a) $k \in Z^+$ that $\zeta (-2k)=0$ b) Use the functional equation and the euler ...