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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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 ...
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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 ...
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1answer
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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 ...
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61 views

Riemann Zeta and Monotonicity

The second paragraph of Wolfram Mathworld Riemann Zeta Function states: The plot above shows the "ridges" of $|\zeta(x+\imath y)|$ for $0<x<1$ and $1<y<100.$ The fact that the ridges ...
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1answer
51 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 ...
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1answer
60 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 ...
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1answer
70 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 ...
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1answer
44 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. ...
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2answers
129 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 ...
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1answer
83 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, ...
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187 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 ...
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1answer
87 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} ...
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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 ...
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1answer
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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). ...
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1answer
67 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 ...
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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 $a,b$ ...
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1answer
269 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 ...
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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 ...
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1answer
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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 ...
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321 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 ...
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2answers
202 views

Riemann Hypothesis and the prime counting function

This article on the prime counting function mentions that 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 ...
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Riemann Hypothesis - prime bounds [duplicate]

Could someone please point me in the direction of a proof of the connection between the bound $\pi(x) = \operatorname{li}(x) + O(\sqrt{x} \log{x})$ and the zeros of the zeta function?
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Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality?

Robin's theorem says that if $$\sigma(n)<e^\gamma n\log\log n$$ holds for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any ...
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hypothesis test and execution? did I do this right?

I am trying to figure this out and I'm not sure if I am doing it right. a) did I select the correct type of hypothesis testing??? b) am I using the numbers in the right place (I have SD for 3.5 for ...
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2answers
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Andre LeClair, Riemann zeta zero approximation?

This sequence A177885 in the oeis seemingly relates imaginary parts of non-trivial Riemann zeta zeros with the LambertW function. The real and imaginary parts of the Riemann zeta function is the sum ...
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Would proof of Legendre's conjecture also prove Riemann's hypothesis?

Legendre's conjecture is that there exists a prime number between $n^2$ and $(n+1)^2$. This has been shown to be very likely using computers, but this is merely a heuristic. I have read that if this ...
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Validity of a functional formula of the Riemann Zeta function across the whole complex plane?

Could someone confirm me the validity of the following formula: $$\zeta\left(z\right)=2^{z}\pi^{z-1}\sin\left(\frac{\pi z}{2}\right)\left(\frac{e^{-\gamma ...
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Riemann hypothesis equivalence statement, where is my error?

I need some feedback on the following: According to the page about the von Mangoldt function at the Mathworld page, the Riemann hypothesis is equivalent to the statement: $$\psi = x + ...
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4answers
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What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
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An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem - If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal ...
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1answer
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Expanding Riemann Zeta

Consider the Riemann Zeta Function $\zeta(x) = 1 + 2^{-x} + 3^{-x} + 4^{-x}...$ Notice the following identity: $a^{-x} = (e^{ln(a)})^{-x} = e^{-xln(a)}$ Therefore: $\zeta(x) = 1 + 2^{-x} + 3^{-x} ...
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Riemann $\zeta(s)$ non-trivial zeros on $Re(s)=1/2$ and the “spin” of the primes? [closed]

Can this conjecture be true? Let me intro before getting to the conjcture, but for those who like to go straight, see the bold paragraph at the end. Recently I was reading through the well known ...
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This is a possible proof of the Riemann Hypothesis [closed]

http://arxiv.org/abs/1305.6845 The above link claims to have solved the Riemann Hypothesis. It's not mine, of course. I just saw this on Tumblr and realized I needed bigger guns. This proof looks like ...
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What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the millenium prize problems, specifically the Riemann hypothesis. I'm not near qualified to even grasp the entire problem, but seeing the hypothesis and the other ...
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Reformulation of riemann zeta

Does this extend to $\mathbb{C}$? $\displaystyle ζ(x) = \int_0^{\infty} \frac{ 1}{\lfloor t\rfloor ^x} dt$, where for $0 \leq t < 1$ we say that $\lfloor t \rfloor = 1$.
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Quantum uncertainty can explain the Riemann Hypothesis?

In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
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A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
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Every prime $p_{n}$ is a prime factor of $\frac{1}{2}$ of all the square-free numbers. [closed]

I have removed my bold claims and the naive question so I can link to this post from another. Edit With each prime, we construct square-free numbers that have that prime as the greatest prime ...
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1answer
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Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list: The Riemann Hypothesis: A Resource ...
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1answer
203 views

Why the Wu-sprung model is not accepted as a solution to Riemann HYpothesis ??

in the Wu-sprung model, given a Hamiltonian in one dimension $$ -y''(x)+f(x)y(x)=E_{n}y(x) \qquad y(0)=0=y(\infty) $$ we can define the function $ f(x) $ implicitly as $$ f^{-1}(x)= 2\sqrt{\pi} ...
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Sums of the negative integer powers of $\zeta$ zeros have an analytical expression…?

The Mathworks page on Riemann's $\zeta$ function says: Let $\rho_k$ denote the $k$th nontrivial zero of $\zeta(s)$, and write the sums of the negative integer powers of such zeros as $$ ...
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2answers
127 views

Plot of a Bessel function if possible

i would like to know where i could find a plot of $$ J_{ia}(2\pi i)$$ (1) using Quantum mechanics i have conjectured that if $ a= \frac{x}{2} $ and $ i= \sqrt{-1} $ then $$ J_{it}(2\pi ...
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A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )

Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number. Odiuos Number Now consider the sequence of functions below $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...