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

15
votes
4answers
1k views

The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n$th square number - it is $n^2$ - but we do not have an exact formula for the $n$th prime number $p_n$! "God may not play ...
0
votes
2answers
54 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
27 views

About questions related to the Riemann's Hypothesis.

Let n be an integer , if n satisfies Robin's Inequality ($\omega(n)$ /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
54 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
50 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
120 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
144 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
39 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 ...
12
votes
2answers
305 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
59 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
52 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
42 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 ...
5
votes
1answer
381 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
votes
2answers
96 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
149 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
113 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
101 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
28 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
65 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
20 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
33 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 ...
0
votes
0answers
71 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 ...
2
votes
1answer
73 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
76 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
76 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
45 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
143 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
90 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
207 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
95 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
99 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 ...
3
votes
1answer
125 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
78 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
112 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
273 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
83 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
167 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
350 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 ...
5
votes
2answers
232 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 ...
1
vote
1answer
77 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 ...
1
vote
0answers
43 views

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 ...
3
votes
2answers
525 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 ...
12
votes
1answer
1k 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 ...
2
votes
0answers
60 views

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 ...
2
votes
0answers
136 views

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 + ...
4
votes
4answers
3k 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 ...
3
votes
0answers
73 views

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 ...
5
votes
1answer
155 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} ...