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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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
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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
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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
56 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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The Riemann Hypothesis The Abc-conjecture and the Twin Primes

Suppose that the Riemann Hypothesis and the abc-conjecture are true. Does it implies an inductive proof of the infinitude of the twin primes since all twin primes lie in the critical line? Following ...
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1answer
41 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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106 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
70 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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2answers
149 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
79 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
91 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). ...
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1answer
62 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
260 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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64 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 ...
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1answer
113 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 ...
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1answer
293 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
158 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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41 views

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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1answer
51 views

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
446 views

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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1answer
942 views

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
2k views

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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142 views

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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2answers
3k views

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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5answers
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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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1answer
83 views

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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431 views

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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2answers
268 views

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
196 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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1answer
92 views

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
115 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 $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...
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If RH is false , could this be true?

Let $\zeta(s)$ be the Riemann zeta function. Assume RH is false , is it possible that we have in the critical strip $\zeta(a_1+ti) = \zeta(a_2+ti) = \zeta(a_3+ti) = \cdots = \zeta(a_n+ti) = 0$ For ...
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3answers
216 views

Riemann Hypothesis: Could there be “simple” ways of getting (partial?) results

Today I did some reading on the Riemann Hypothesis and decided to play around with $\zeta(s)$ a little bit. (In case my question is ridiculous, I'm a student who has no experience dealing with zeta ...
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1answer
188 views

A follow-up problem once the Riemann hypothesis has considered proven to be truth? [closed]

What will happen if a mathmatician have prove that 99.999....% of the solution stay on the critical line and receive the prize but after that another mathmatician find finite numbers of interesting ...
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1answer
275 views

Riemann hypothesis and diophantine equation

I read that showing Riemann hypothesis is true was equivalent to showing a particular diophantine equation doesn't have any solution. Is there an explicit example of such a diophantine equation?
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181 views

Is $M(x)=O(x^σ)$ possible with $σ≤1$ even if the Riemann hypothesis is false?

The wiki page on Mertens conjecture and the Connection to the Riemann hypothesis says Using the Mellin inversion theorem we now can express $M$ in terms of 1/ζ as $$ M(x) = \frac{1}{2 \pi i} ...
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292 views

Is classifying one dimensional generalized quasicrystals worthwhile strategy to approach RH?

Works done: After fruitlessly poring over books on zeta functions, it seems Freeman Dyson's sotto voce nudge to classify generalized one-dimensional quasicrystals is a way to go. As he writes: ...
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613 views

$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions

In a paper about Prime Number Races, I found the following (page 14 and 19): This formula, while widely believed to be correct, has not yet been proved. $$ \frac{\int\limits_2^x{\frac{dt}{\ln ...
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2answers
717 views

Are there examples that suggest the Riemann Hypothesis might be false?

Are there examples that might suggest the Riemann hypothesis is false? I mean, is there a zeta function $ \zeta (s,X) $ for some mathematical object $X$ with the properties $ \zeta (1-s,X) $ and ...
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271 views

Euler's Choice and Riemann's Oversight?

Euler's Choice: When Euler crafted the zeta function, he knew that $\zeta(1)$ diverged, so he made $\zeta(1)$ undefined. When he crafted the zeta generating function using the Bernoulli numbers, ...