# Tagged Questions

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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### Errors in following paper

In the paper "On the Level Curves of the Xi Function" http://arxiv.org/abs/1002.0352v8, John Breslaw takes a very similar approach to a study of the Riemann hypothesis I did while I was in my first ...
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### Is this a valid attempt at the Riemann Hypothesis? [closed]

From Marcus Du Sautoy's book “The music of the primes”, there is a method of finding a very long list of N consecutive numbers which are not primes. e.g $101!+2, 101!+3,...,101!+101$ all of which are ...
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### Titchmarsh S function

SO it is known that Titchmarsh S function $$S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of *riemann hypothesis * gives $$S(T)=O(\frac{\log T}{\log \log T})$$ can ...
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### RH would follow from $\displaystyle \frac{p_{n+1}}{p_{n+1}-1}<\frac{\log\log N_{n+1}}{\log\log N_n}$ for all $n>1$; what is my mistake?

Let $N_n=\prod_{k=1}^np_k$ be the primorial of order $n$,$\gamma$ be the Euler-Mascheroni constant and $\varphi$ denote the Euler phi function. Nicolas showed that if the Riemann Hypothesis is true, ...
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### Some clarifications on analytic continuation of Riemann's Zeta function on $\frac 1 2$

Here's my problem: Riemann's Zeta function converge iff $x>1$ so if I want to have a finite value for $\zeta(\frac 1 2)$ I need to use it's analytic continuation but Riemann's hypothesis states ...
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### For Riemann Hypothesis, many people seek physics intuition, why not for Goldbach Conjecture ?

All: As we all know, for Riemann Hypothesis research, many people seek physics intuition, to understand more fundamental reasons why Riemann Hypothesis shall hold. In this direction, we have ...
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### Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
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### Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?

I realize that asking this question is like presenting to a patent attorney a wheel-less skateboard while asking to patent a hoverboard. Anyways. Lagarias version of the Riemann hypothesis sets a ...
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### Why proving Riemann hypothesis is practically important?

I agree that studying pure mathematics is meaningful by intellectual curiosity itself. However, after AKS algorithm is found, I have a question "Is still Riemann hypothesis practically important ...
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### Riemann Hypothesis, is this statement equivalent to Mertens function statement?

All: I saw one form of Riemann Hypothesis, it says: $$\lim ∑(μ(n))/n^σ$$ Converges for all σ > ½ Is this statement same as the order of Mertens function is less than square root of n ?
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### parity problems for sieve methods, is it only for Selberg Sieve or for all sieve methods?

It is said that sieve methods have parity problems. Terence Tao gave this "rough" statement of the problem: "Parity problem. If A is a set whose elements are all products of an odd number of primes ...
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### How to prove that Riemann zeta function is zero for negative even numbers? [duplicate]

Can anyone please explain to me how to prove that Riemann zeta function is 0 for all negative even numbers. In many references , they have just given the statement without any proof. Any explanation ...
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### What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
371 views

### Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
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### What is wrong with this argument (RH)?

I should be very grateful if someone would point out the error in the following argument, since it seems too trivial to be valid: Let $\{pp_{n+1},pp_{n}\}$ denote the interval between prime powers, ...
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### Prime number distribution theory for dummies

For the distribution of prime numbers there is a hypothesis which predicts the possible positions of prime numbers called Riemann hypothesis http://en.wikipedia.org/wiki/Riemann_hypothesis#...
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Could Riemann' Hypothesis be proven true using Robin's Inequality and that a counter-example to Riemann's Hypothesis can not have a divisor that is a prime number to the exponent 5 ,according to some ...
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### If Lagarias' inequality is wrong, are there infinitely many counterexamples to it?

I do know that since Robin's (RI) and Lagarias' (LI) inequalities are both equivalent to RH, they're also equivalent one another, hence if RI is false, so is LI. And Robin proved there are infinitely ...
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### Usage of Complex Numbers in the Riemann Hypothesis.

I don't have a very good understanding of the Riemann Hypothesis, however that being said, could someone explain to me why complex numbers are used, instead of just using real numbers? Everything I've ...
573 views

### Why are people more interested in the Riemann hypothesis than Goldbach's conjecture? [closed]

One of my friends, a math professor, told me almost every one of his colleagues (in the math department) had attempted to prove the Riemann hypothesis at some point in their life (maybe secretly). ...
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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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### Identity for frequency of integers with smallest prime(n) divisor

An identity for A038110 and A038111: $$\frac{\phi(e^{\psi(p_{n}-1)})}{e^{\psi(p_{n})}}=\frac{\prod _k^{n-1} \left(1-\frac{1}{p_k}\right)}{p_n},$$ where $\psi(\cdot)$ is the second Chebyshev function ...
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### If one wanted to study the Riemann Hypothesis, what should they study?

I've seen posts of a similar nature that list numerous books and papers about the Riemann Hypothesis. But, assuming one has no knowledge of the subject, where should they start studying?
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### Is Riemann Hypothesis provable?

Loosely speaking, there are three kinds of propositions. Those propositions which are true and can be proved to be true. Those propositions which are false and which can be proved to be false....
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### Category theoretic approaches to Riemann Hypothesis?

I was wondering if there has been any category theoretic advancements in the study of the Riemann Hypothesis and the theory surrounding it? This question is meant in the same vein as these questions,...
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### (Easy?) consequence of the Riemann Hypothesis

I'm trying to show that the relation $\psi(x)=x+O(\sqrt{x}\log ^2 x)$ (consequence of the Riemann hypothesis) implies $\pi(x)=Li(x)+O(\sqrt{x}\log x)$, where $Li(x)=\int_2^x \frac{dt}{\log t}$. I ...
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### Can you find such a function that satisfies the RH statement?

For instance, see Generalized Riemann Hypothesis. It conjectures that if $L(\chi, s) = 0$, and $0 \leq \Re(s)\leq 1$, then $\Re(s) = 1/2$. Then is there a function $f(s)$ that you can think of that ...
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### Consequence of the Riemann Hypothesis

So I watched this video: http://m.youtube.com/watch?v=rGo2hsoJSbo And it included the fact that a consequence of RH is that there will always be a prime number between consecutive cubic numbers. I ...
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### Periodicity in Riemann zeros.

Has someone studied if the non-trivial zeroes of the Riemann zeta function has some "periodicity" or "quasiperiodicity"? And what about generalized zeta functions and/or L-functions?
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### Probability and discrete mathematics

Has the presumption of the Riemann Hypothesis had any impact on probability? ie: Are there any important theorems in probability that begin with "Give that the RH is true..." (It seems likely, given ...
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### About the riemann's hypothesis .

Robin's inequality and Lagarius's inequality (forgive spelling) are two simpler statements that are equivalent to Riemann's hypothesis ;are there any others?
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### Riemann Hypothesis Proof - legit? [closed]

https://www.academia.edu/6829966/Proof_the_Riemann_Hypothesis_Cite_This_Article The above paper claims to prove the Riemann Hypothesis, but seems somewhat suspect in its length, broken English, and ...
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 ...
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 ...