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.

learn more… | top users | synonyms

0
votes
0answers
59 views

The P versus NP and The Riemann Hypothesis [on hold]

This question is related to this article UPDATED which states Suppose that the Riemann Hypothesis is true such that For all $n>1$ and prime $p$ there exists a complex number ...
0
votes
0answers
28 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 ...
0
votes
1answer
50 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 RH 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 ...
-2
votes
2answers
131 views

Why have so many people ignored the recent proof of the Riemann hypothesis? [closed]

The Riemann Hypothesis was proved 2 May 2013 by Jin Gyu Lee is his paper Zeros of the Riemann Zeta Function i.e. arXiv:1305.0323. This proof has been verified as the real deal. The proof is simple and ...
3
votes
0answers
39 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, ...
13
votes
1answer
107 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
votes
1answer
45 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
votes
2answers
1k views

Can you solve this captcha?

I found the following problem in a captcha: (and I was really surprised) What does it mean, and what would the solution be? EDIT: It looks, from comments and answers, that this is a consequence ...
0
votes
0answers
30 views

About Robin's inequality and his work on counterexamples to the inequality.

Did Robin's work on his inequality and its relation to Riemann's Hypothesis prove that any counterexample to the inequality could not have a prime divisor with an exponent of 5 or more? If so how did ...
1
vote
2answers
68 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.
47
votes
7answers
5k views

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 $n^\text{th}$ ...
0
votes
2answers
66 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?
0
votes
0answers
42 views

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. ...
1
vote
1answer
71 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$$\ ...
1
vote
0answers
64 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, ...
0
votes
0answers
151 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 ...
4
votes
1answer
165 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 ...
0
votes
0answers
45 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 ...
7
votes
4answers
504 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 ...
13
votes
2answers
350 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). ...
1
vote
0answers
69 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, ...
2
votes
0answers
56 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\ ...
1
vote
0answers
48 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
votes
1answer
401 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 ...
1
vote
2answers
137 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?
2
votes
0answers
222 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 ...
5
votes
1answer
118 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 ...
7
votes
1answer
116 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 ...
0
votes
1answer
32 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 ...
2
votes
0answers
70 views

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 ...
1
vote
0answers
23 views

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?
1
vote
1answer
36 views

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 ...
0
votes
0answers
38 views

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?
5
votes
2answers
1k views

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 ...
2
votes
1answer
94 views

Error term of the prime number theorem in arithmetic progressions

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 ...
1
vote
1answer
85 views

Prime Counting: Relationship between Chebyshev's function and the Prime counting function

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 ...
1
vote
1answer
80 views

If Cramér's is proved?

Harald Cramér proved that under this assumption that the Riemann hypothesis is true., the gap $g_n$ satisfies $$g_n = O(\sqrt{p_n} \ln p_n) ,$$ using the big O notation. Later, he conjectured that ...
0
votes
1answer
48 views

A question about an asymptotic formula

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. ...
4
votes
2answers
151 views

What would be the consequences of proving Riemann's hypothesis for Legendre's conjecture?

I've heard somewhere that Riemann's hypothesis doesn't imply Legendre's conjecture. But if Riemann's hypothesis is true, would an interval maybe a bit larger than $[n^2,(n+1)^2]$ contain always at ...
1
vote
1answer
98 views

Non-trivial zeros off critical line

If non-trivial zeros lay off the critical line (as shown in the picture below), would they have to come in fours rather than conjugate pairs (as the diagram shows)? I am presuming they would, ...
4
votes
2answers
227 views

Proving the falsity of the Riemann Hypothesis

The Riemann Hypothesis is equivalent to the statement: $$|\pi(x)-{\rm li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657,\text{ (Schoenfeld, 1976)} $$ Which can be visually ...
2
votes
1answer
104 views

Analytic Continuation of Riemann Zeta Function

How do we show that this holds for $\operatorname{Re}(z)>0$ (and not $1$) $$\zeta(z)= \sum_{i=0}^{m-1} n^{-i} + \frac{m^{-z}}2 +\frac{m^{1-z}}{z-1} -z\int_{m}^{\infty} ...
5
votes
0answers
111 views

Mertens conjecture & Riemann hypothesis [closed]

The Mathworld page on the Mertens Conjecture states that $$\limsup_{n\rightarrow\infty}|M(n)|n^{-1/2}=\infty$$ seems very probable (Odlyzko and te Riele 1985). Would it not then follow that ...
4
votes
1answer
146 views

The relationship between Golbach's Conjecture and the Riemann Hypothesis

My question pertains to two famous groups of related conjectures: Goldbach's Conjecture (GC); Goldbach's Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH). ...
0
votes
1answer
83 views

Application of the Robins Equality

The Robin's inequality says - If the Riemann hypothesis is true then - $$\sigma(n) < e^{\gamma}n \log(\log(n))$$ holds true for all $n \in \mathbb{N}$ Now it is proved for all $5-$ free integers ...
1
vote
0answers
114 views

About $f(s)=\sum_{a^2+b^2>0} \frac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}=0$ and the Extended Riemann Hypothesis.

Let $s$ be a complex number with a strictly positive real part ($Re(s)>0$). Let $f(s)=\sum_{a^2+b^2>0} \dfrac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}$ where the sum runs over all positive integers $a,b$ ...
1
vote
1answer
274 views

Can we derive this from Mertens theorem?

From "Elementary Methods in Number theory" by 'Melvyn B. Nathanson', I know from Merten's first theorem that $\displaystyle R(x)=\sum_{p\leq x}\dfrac{\ln p}p-\ln x=O(1)$ but can it be ...
1
vote
0answers
84 views

Can the error term involved in the PNT be expressed in a Galois theoretic framework?

According to Wikipedia, the current best error term for the prime number theorem is $\pi(x)-\mathrm{Li}(x)=O\left(x\exp\left(-\frac{A(\ln x)^{3/5}}{(\ln\ln x)^{1/5}}\right)\right)$, while RH is ...
1
vote
1answer
200 views

Have all the zeros of the Riemann Zeta function real part smaller than 1?

I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$ have real part ...
18
votes
1answer
357 views

Is there a good (preferably comprehensive) list of which conjectures imply the Riemann Hypothesis?

I wanted to prepare a presentation for the students I tutor on the Clay Millennium problems. This is directed at the Riemann Hypothesis and the Generalized Riemann Hypothesis. The Wikipedia article ...