Questions tagged [riemann-hypothesis]

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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7
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0answers
88 views

Bounds for the $n^{th}$-prime number and the inverse logarithmic integral?

Are there any bounds for the $n^{th}$-prime number $p(n)$ and the inverse logarithmic integral $\text{ali}(n)=\text{li}^{-1}(n)$ under the Riemann Hypothesis? $$|p(n)-\text{ali}(n)|<?$$ here is ...
0
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0answers
37 views

If $ \sum_{n=1}^{\infty} a_n n^h=\sum_{n=1}^{\infty}a_nn^{-h}=0$ for nonzero $a_i$, is there a way to show $h=0$ is the unique solution? [closed]

If I have $$ \sum_{n=1}^{\infty} a_n n^h = 0\quad\mathrm{and}\quad\sum_{n=1}^{\infty} a_n n^{-h} = 0\\ \mathrm{with}\quad a_1,a_2,...\neq0\\ a_n = \frac{(-1)^{n}}{\sqrt{n}}\sin(t \ln(n)) $$ Is there a ...
0
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1answer
85 views

Is the value of the following limit with an infinite number of variables equal to zero or is it not decidable?

According to the alternating series test a general term $b_n$ in the sum: $\sum\limits_{n=1}^{\infty}(-1)^{n+1}b_n$ should converge to zero: $\lim\limits_{n \rightarrow \infty} b_n = 0$ From reuns ...
27
votes
0answers
554 views

Colossally abundant numbers and the Riemann hypothesis

A recent tweet by John Baez has reminded me of the astonishing fact$^1$ that the Riemann hypothesis (RH) can be disproved by finding a number $n > 5040$ such that $$\frac{\sigma(n)}{n \ln\ln n} >...
1
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0answers
52 views

How do you turn an upper bound of the second Chebyshev function into a corresponding lower bound?

In Table 5.1 on page 103 of the book The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators by Machiel van Frankenhuijsen the author states: $$\textit{Riemann hypothesis} \iff ...
11
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3answers
385 views

I am looking for class of math problems which are provable in ZF if and only if they are provable in ZFC

I know that P vs NP and Riemannian hypothesis are of this class but could not find any article on that. I would also appriciate links or books on related theme. My question is: what are some other ...
0
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0answers
60 views

How come $\zeta(s)$ and $\zeta(s)=(1-2^{1-s})^{-1}\eta(s)$ looks different?

In this question, I have asked How I can plot the critical line of $\zeta(s)$ as the definition of $\zeta (s)$ is valid only for $s\geq 1$. One way to plot is to use $\eta(s)$ $$\eta (s)=(1-2^{1-s})\...
2
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1answer
126 views

Non trivial zeros of Riemann zeta function

The non trivial zeros of Riemann zeta function , x$\zeta(s)$ lies in the critical strip $0<\Re(s)<1$ Riemann Hypothesis states that all the zeros of Riemann zeta function, $\zeta(s)$ lies on ...
1
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0answers
30 views

Is there a bound of $\Delta(x)$ related to Riemann prime-counting funtion?

$$\Delta (x)=\left(\pi (x)-\operatorname {R} (x)+{\frac {1}{\ln x}}-{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}\right){\frac {\ln x}{\sqrt {x}}},$$ where $\operatorname{R}(x)$ is Riemann R function....
0
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1answer
45 views

Question Regarding Relationship between the Riemann Zeta Function and the Dirichlet Eta Function

So I was looking at the Riemann hypothesis and I saw the relationship between the Riemann zeta function and the Dirichlet eta function which really confused me because I didn't understand how a ...
0
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2answers
78 views

Pustyl’nikov’s equivalent criteria for RH

In this paper (Pustyl’snikov 1999 Russ. Math. Surv. 54 262), Pustyl’nikov proved the following two theorems Theorem 1 All the even derivative of $\xi(s) $ at point $s=1/2$ are strictly positive. ...
0
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1answer
174 views

A disproof of RH? [closed]

Does this paper of Sondow et. al, in which they propose a disproof of RH, have reasonable arguments ? The Riemann Hypothesis is not true
1
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1answer
78 views

Does the Mandelbrot Set with a power of -1 produce a Farey Sequence, and if so, can this be connected to the Riemann Hypothesis?

I'm by no means an expert in Farey Sequences nor understanding the Riemann Hypothesis - but I do know there is some relationship between them, as alluded to in this question. As such, if the ...
0
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1answer
88 views

$\sum_{\rho}\frac{1}{|\rho|^2}=\sum_\rho\frac{1}{\rho(1-\rho)}$

Riemann Hypothesis(RH) is equivalent to the statement " https://mathoverflow.net/questions/91280/is-this-sum-of-reciprocals-of-zeta-zeros-correct "$$\sum_{\rho}\frac{1}{|\rho|^2}=\sum_\rho\...
4
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1answer
141 views

Does this disprove the Riemann Hypothesis?

I found interesting results when I put in certain values for z into zeta(z). I believe my findings do not disprove the Riemann hypothesis, but I am not so sure. When I input z = 1/3 + i into the ...
0
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0answers
43 views

What happens if you compose the zeta function with two circle inversion mappings in this specific way?

Let $I^a_b : \mathbb{C} \to \mathbb{C}$ be defined as circle inversion mapping of a circle at the point $a \in \mathbb{C}$ of radius $b = \frac{1}{r}$ What happens to the zero's of the zeta function ...
-1
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1answer
303 views

$\int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt$ where $\zeta$ is the Riemann zeta function

$$I=\int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt$$ where $\zeta$ is the Riemann zeta function. It is known by Balazard Saias and Yor Paper that I is integrable and $0\...
2
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1answer
123 views

$f(z)=\frac{z}{1-z} \zeta(\frac {1}{1-z})$ belongs to Hardy Space $H^{1/3}(\mathbf{D})$

The Hardy space $H^p(\mathbf{D})$ is the vector space of holomorphic functions $f$ on the open unit disk that satisfy: $$ \sup_{0< r<1}\left(\frac{1}{2\pi} \int_0^{2\pi}\left|f \left (re^{i\...
0
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1answer
54 views

Prove$\sum_{|\alpha|<1, f(\alpha)=0}\log \frac{1}{|\alpha|}= \sum_{\Re(\rho)>1/2}\log\ |\frac{\rho}{1-\rho}|$ [closed]

Let $$ f(z)= (s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta$ denotes the Riemann Zeta function. Prove that,$$\sum_{|\alpha|<1, f(\alpha)=0}\log \frac{1}{|\alpha|}= \sum_{\Re(\rho)>1/2}\log\ ...
2
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1answer
80 views

Absolute convergence of a double matrix and of $\frac{1}{\zeta(\frac{1}{2}+\epsilon)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{\frac{1}{2}+\epsilon}}$

According to Wikipedia the convergence of the right hand side of: $$\frac{1}{\zeta(\frac{1}{2}+\epsilon)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{(\frac{1}{2}+\epsilon)}}$$ for $\epsilon$ an arbitrary ...
4
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2answers
100 views

Density and distributions of those numerically or analytically KNOWN solutions of Riemann $\zeta(1/2 + r i)=0?$

We know the conjecture about the Riemann hypothesis is about the nontrivial zeros are on $$(1/2 + r i)$$ for some $r \in \mathbb{R}$ of Riemann zeta function. My question is how much is known about ...
6
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1answer
180 views

The Riemann hypothesis as a statement about natural numbers

In this answer it is claimed that the Riemann hypothesis can be expressed as a statement about natural numbers. How would that look like?
0
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1answer
142 views

$ \int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t$ is Convergent

Define $$I= \int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t.$$ Balazard, Saias and Yor showed that the Riemann Hypothesis is equivalent to the statement that $I=...
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1answer
116 views

Prove that $\Re(\rho)=1/2$

Given $$\frac{\mid \int_{x=1}^\infty(\frac{1-x+[x]}{x^2})x^{1-\bar{\rho}} dx\mid}{ \mid \int_{x=1}^\infty(\frac{1-x+[x]}{x^2})x^\rho dx\mid }=1$$ where [x] denotes the greatest integer function and $...
0
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1answer
57 views

Prove $\int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x} $ is Entire

Define $$f(s)= \int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x} $$ where $ \psi(x)= \sum_{n=1}^{\infty}e^{-n^2\pi x}$ is the Jacobi Theta function. Claim- $f(s)$ is Entire My Try :- ...
8
votes
1answer
422 views

Why is proving the Riemann Hypothesis so hard?

The Riemann Hypothesis is considered by many to be the most important unsolved problem in pure mathematics. Several attempts have been made in the last 150 years (here some of them are reported). RH ...
0
votes
1answer
243 views

Is it hopeless to try and solve this equation analytically?

Can this equation be solved with analytical methods, or is it only numeric methods since current mathematical tools don't go that far? Its complex roots are the same as the roots of the zeta function ...
1
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0answers
48 views

Direct derivation of $J(x) = \operatorname{li}(x) - \sum_{\rho} \operatorname{li}(x^\rho) + \int_{x}^{\infty} \frac{dt}{t(t^2-1)\log t} -\log 2 $

I'm reading the Riemann's article on the number of prime numbers with a detailed explanation by W. Dittrich, and stucked to prove the Riemann's final formula $$ J(x) := \sum_{p^n \leq x} \dfrac{1}{n} =...
2
votes
2answers
256 views

If $z_n$ are the zeros of the zeta function, what is the limit of $\Im{(z_n)}$ as $n$ goes to infinity?

Sorry if this question has already been asked, but it's a little difficult to look things up in Google if the statement of the problem is not very simple and involves symbols that Google doesn't ...
1
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0answers
69 views

Solutions to $\frac{1}{1-\frac{A}{B}}+s=\left(\frac{1}{1-\frac{B}{A}}-s\right)^*$ over the Complexes.

Here I describe how I arrived at this equality, or its starting point: $$\lim_{n\to \infty } \, \left(\frac{1}{1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s-\frac{...
1
vote
1answer
74 views

Reference for another equivalent of Riemann hypothesis

I don't know why but i assumed that $\ln \theta(x) + E < \sum \limits_{p \leq x} \frac{\ln p}{p} $ for all real numbers $x \geq 2$ and with $E = -\gamma - \sum \limits_{P} \frac{\ln p}{p(p-1)} \...
0
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0answers
54 views

Has anyone had studied the Riemann hypothesis with non-standard models?

In symbolic logic, non-standard numbers is common nomenclature for a model of numbers which satisfies the axioms of some arithmetical system, but doesn't consist of what people would normally think of ...
1
vote
0answers
87 views

How many counterexamples the Riemann Hypothesis, if false, can have?

Are there papers, which show, that there can not be infinite many counter-examples for the Riemann Hypothesis? Can you give me references for these papers? I enjoy thinking and learning. I mean no ...
0
votes
1answer
49 views

Difference in Robin's inequality assuming R.H.

If we assumed R.H. what is the best upper/lower bounds for $ E +\ln \theta(x) - \sum \limits_{p \leq x} \frac{\ln p}{p}$ And if one can also give the best upper/lower bounds for $ E +\ln x + (\frac{\...
-1
votes
1answer
113 views

Find $\sum_{\rho}\frac{1}{\rho^3}$ where $\rho$ is a non trivial zero of zeta function

I am reading the Equivalents of Riemann Hypothesis: Arithmetic Equivalents pg. 35 Lemma 2.10. If $\rho$ is a non trivial zero of the Riemann Zeta function then, $\sum_{\rho}\frac{1}{\rho}=1+\frac{\...
0
votes
1answer
50 views

Exact value of a sequence of coefficients.

Let $a_1=-1$ and $a_n=-1-\sum_{j=1}^{n-1}a_j[n/j]$, where $[x]$ is the integer part of $x$. Is it always true that $a_n\in\{-1,0,1\}$? These coefficients are determined by the formal equality $\chi_{(...
0
votes
0answers
46 views

Is there any closed form for riemenn zeta function for real domain?

I wonder if there is any closed form for the riemann zeta function at real domain (eaither including 1 or not)? So at only real values of s, does the zeta function have a closed form? I would be ...
2
votes
1answer
89 views

$\sum_{\Im(\rho)>0}\frac{1}{\mid{\rho-\frac{1}{2}}\mid^2}\ll \int_{1}^{\infty} \frac{d(t\log t)}{t^2}$.

In the book Equivalents of Riemann Hypothesis Kevin Broughan Volume 1 , pg 38, Riemann Xi function is defined as $\xi(s)=\xi(0)\prod_{\Im(\rho)>0}(1-\frac{s(1-s)}{\rho(1-\rho)})$. Then he says that ...
-2
votes
1answer
138 views

How Dr. Zhu work on proving Riemann's Hypothesis was reviewed by the mathematical community? [closed]

I am not aware of the logical mistakes or flaws of the Dr. Zhu paper: Zhu Y., The probability of Riemann's hypothesis being true is equal to 1, arXiv:1609.07555 (2016, 2018). https://arxiv.org/abs/...
0
votes
0answers
112 views

Non-trivial zeros of Davenport-Heilbronn function

From the book The Riemann Zeta-Function by Karatsuba we know that for Davenport-Heilbronn like function equations, there are many non-trivial zeros off the critical line. However I found this paper on ...
0
votes
0answers
106 views

zeros of the Riemann Hypothesis

I watched a video explaining that using the non-trivial zeros of the Riemann zeta function, we can approximate the prime counting function to have a pretty high accuracy I would like to know, how many ...
3
votes
0answers
97 views

Distribution of non-trivial zeros of the Riemann ζ‑function

This question is about statistical properties of the distribution of the complex part of non-trivial zeros $\rho_n$ of the Riemann $\zeta$‑function. The zeros tend to become more dense as $n$ grows, ...
1
vote
0answers
87 views

$\prod_\rho (1-\frac{s}{\rho})$=$\prod_{\Im(\rho)>0 }(1-\frac{s}{\rho})(1-\frac{s}{1-{\rho}})$

I am reading H.M. Edwards Riemann zeta function and in page 42 section 2.5 he writes these 2 products equal. I am not able to understand. $\prod_\rho (1-\frac{s}{\rho})$=$\prod_{\Im(\rho)>0 }(1-\...
1
vote
0answers
37 views

Bounding summation

Let $T$ be a sufficiently large real number and $\gamma$ be imaginary part of non-trivial zeros of Riemann zeta function. I would like to bound the sum like $$\sum_{0<\gamma\leq T}\gamma^{-1/4}.$$ ...
4
votes
0answers
75 views

Inequality for divisor sigma $\sigma_{\nu}(n)$ function.

The divisor sigma function is defined as $$ \sigma_{\nu}(n):=\sum_{d|n}d^{\nu}\textrm{, }n\in\textbf{N} $$ It is known that under Riemann's hypothesis it holds the following inequality due to ...
1
vote
1answer
82 views

Calculate $\int_1^\infty \frac{ (1-x+[x])(x^{1-\sigma}-x^\sigma)cos(ln x)}{x^2}dx$

Calculate $\int_1^\infty \frac{ (1-x+[x])(x^{1-\sigma}-x^\sigma)cos(ln x)}{x^2}dx $ where [.] denotes the greatest integer function and $0<\sigma<1$. My try - $\int_1^\infty \frac{ (1-...
1
vote
0answers
74 views

Conjectures/Hypotheses which are stronger than Riemann hypothesis

I would like to ask what conjectures or hypotheses are present which are stronger than the original Riemann hypothesis i.e, if proven imply that the RH is true/false. One example would be the Merten's ...
3
votes
1answer
105 views

Find $\sum_{n=1}^{\infty} \frac{n^{\sigma -1} (n+\sigma )-(n+1)^{\sigma }}{\sigma(1-\sigma)}$ for $ 0<\sigma<1$

Find $\sum_{n=1}^{\infty} \frac{n^{\sigma -1} (n+\sigma )-(n+1)^{\sigma }}{\sigma(1-\sigma)}$ for $ 0<\sigma<1$ My try $ \sum_{n=1}^{\infty} \frac{n^{\sigma -1} (n+\sigma )-(n+1)^{\sigma }}{\...
2
votes
0answers
54 views

Are there any known rigid properties of the counterexample zeroes to RH assuming the failure of RH?

One often suggested "route" for the Riemann Hypothesis is through the study of quasicrystals. In particular, assuming RH, the nontrivial zeroes form a 1-dimensional quasicrystal. If we can ...
3
votes
0answers
70 views

Proving Lindelof hypothesis using Phragmen-Lindelof theorem and Riemann Hypothesis

Apparently, assuming the Riemann Hypothesis to be true, we could use Phragmen-Lindelof theorem to deduce the Lindelof Hypothesis. I searched for a while but could not find a proof. Could anyone share ...

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