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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1answer
240 views

Riemann hypothesis reformulation - again

Yesterday I started to write a paper about the reformulation of the Riemann Hypothesis. My idea was to map the function such that all of the trivial zeros are outside of the unit disk, and the ...
3
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2answers
62 views

What are the most promising approches for solving Riemann hypothesis? [closed]

I'm not a mathematician but still I'm very interested in Riemann hypothesis. I discovered it with the Numberphile channel. I would like to know what are the current work done of this subject and if ...
2
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1answer
29 views

Bibilography: Riemann's hypothesis and positive semi-definite billinear forms

This is a bibliography request: I remember browsing through a book, some years ago, in a library, in which Riemann's hypothesis was proved over some type of fields (I cannot remember what type), the ...
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0answers
32 views

What is general Riemann's Hypothesis? [duplicate]

What makes it so important in analytic number theory?
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68 views

Can $\zeta(s)$ be written in the form $\zeta(s)=\Re(\zeta(s))+i·\Im(\zeta(s)) $ for some subset of $\mathbb{C}$?

Can $\zeta(s)$ be written in the form $\zeta(s)=f(s)+g(s) i $ for some subset of $\mathbb{C}$? I mean, is it possible to develop at least one of the formulas of $\zeta(s)$ so you get something like... ...
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2answers
120 views

Would the Riemann Hypothesis being false affect how frequently primes occur in the number system?

I want to know that if Riemann hypothesis is false (big assumption) would that lead to any effect in how frequently primes occur . Well I got this half cooked information from here: ...
7
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1answer
125 views

Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
-2
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1answer
47 views

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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2answers
2k views

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 ...
2
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0answers
45 views

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 ...
6
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2answers
270 views

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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1answer
59 views

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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0answers
79 views

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 ...
3
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0answers
76 views

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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0answers
194 views

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 ...
5
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1answer
104 views

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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0answers
39 views

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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1answer
98 views

How to prove that Riemann zeta function is zero for negative even numbers?

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 ...
9
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2answers
294 views

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 ...
8
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2answers
255 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 ...
8
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0answers
198 views

Are these known telescoping series for $\zeta\left(\frac12\right)$?

There are many known telescoping series for $\zeta(s)$ and I was playing with the following two: $$\displaystyle \zeta(s) = \frac{1}{(s-1)} \left(\sum _{n=1}^{\infty } \left( {\frac {n}{(n+1)^{s}}} - ...
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0answers
113 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
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2answers
117 views

References for Riemann Hypotheis giving the best bound for Prime Number Theorem

Which books cover the proof that Riemann Hypothesis is equivalent to the best error bound for the Prime Number Theorem? My understanding is that Riemann Hypothesis is equivalent to the best bound of ...
1
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1answer
38 views

big $\mathcal O$ for number of prime in an interval?

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function $$\pi(x)=Li(x)+\mathcal O(\sqrt x \log x)$$ I am trying to understand how to deal with the ...
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0answers
53 views

Koch's version of the Riemann hypothesis for $x=p^2$

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function $$\pi(x)=Li(x)+O(\sqrt x \log x)$$ For this equation, does there exist any reference or does ...
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2answers
216 views

Is this claim true$(\xi \circ k)(s)=(k \circ \xi )(s)=0$ $\implies$ $k(s)=\zeta(s)=0 $ is true if and only if the Riemann Hypothesis is false?

It is well known that $\xi(s)=\xi(1-s)$ is a verified functional equation for all complex $s$, where $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. let $k(s)=\xi(1-s)$ and ...
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0answers
85 views

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
14
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1answer
179 views

Consequences of the negation of the Riemann hypothesis

There are many sources documenting the consequences of the Riemann hypothesis, but I can't find one discussing the consequences of its negation, particularly concerning the prime distribution. Can ...
0
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1answer
59 views

Is it an open problem about Riemman Hypothesis non-trivial zero? [duplicate]

Let's assume RH was correct, and $1/2+Ki$ is any one of non-trivial zero of $\zeta$, is following problem open? 1) $K$ is irrational number 2) $K$ is transcendental number
9
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2answers
1k views

Can you solve this captcha?

I found the following problem in a captcha: (and I was really surprised, I expected just regular blurred or distorted text) What does it mean, and what would the solution be? EDIT: It looks, ...
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2answers
79 views

About the 'sigma' function.

Is it true that if $n$ divides $m$ , $\sigma(\frac mn) \leq \frac{\sigma(m)}n$. If so this has a bearing on counterexamples to Robin's inequality.
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8answers
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The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n^\text{th}$ square number – it is $n^2$ – but we do not have a (useful) exact formula for the $n^\text{th}$ ...
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2answers
78 views

Regarding the $\sigma (n)$ function.

This question relates to Robin's Inequality. Is $\sigma{(n^2)}$ < (2 n) $\sigma{(n)}$ ? For what integer values of n is this satisfied?
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About questions related to the Riemann's Hypothesis.

Let n be an integer , if n satisfies Robin's Inequality ($\sigma(n)$ /n < $e^{\gamma}$ lnlnn) say n is 'regular'. If n doesn't satisfy Robin's Inequality say n is a 'counter' or a counter-example. ...
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1answer
89 views

If Robin's inequality ever fails, are there only finitely many colossally abundant numbers that satisfy it?

Let$\ \sigma(n)$ be the sum-of divisors function, with the divisors raised to$\ 1$. If the Riemann Hypothesis is false, Robin proved there are infinitely many counterexamples to the inequality$$\ ...
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0answers
66 views

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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232 views

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 ...
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1answer
215 views

About Riemann's Hypothesis.

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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0answers
60 views

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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4answers
554 views

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 ...
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2answers
462 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). ...
2
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0answers
84 views

Similarity of two limits related to the sum of divisors $\sigma(n)$ and the harmonic numbers $H_n$

Given that the sum of divisors has the form: $$\large \sigma(n) = \sum _{k=1}^n \lim_{s\to 0} \, \left(\frac{(s+1) (-1)^{\frac{2 n}{k}}+s-1}{k \cdot s \cdot 2}\right)^{-1}$$ $$1, 3, 4, 7, 6, 12, 8, ...
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0answers
60 views

Prime bounds under RH

Continuing from here, since $$ \sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}=\operatorname{li}(n)-\sum_{k=1}^{\infty}2\ ...
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55 views

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 ...
6
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1answer
431 views

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 ...
2
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2answers
182 views

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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0answers
329 views

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 ...
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1answer
156 views

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 ...
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1answer
178 views

(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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1answer
34 views

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 ...