# 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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### Count the number of ordered triples of positive integers whose product is not greater than a given number?

Given N, count the number of ordered triplets(a,b,c) whose product $abc \leq N$. I have found the series here . But I am not sure I understand Benoit Cloitre work which proposed an efficent way to ...
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Consider the summatory function $$f(x)=\sum\limits_{n=1}^x 1_{n\ne k^m}\tag{1}$$ where $1_{n\ne k^m}$ is the non-integer-power indicator function which returns $1$ when $n$ is a non-integer-power and $... 2 votes 1 answer 67 views ### Why$\sum_{\gamma>0} (\frac{\sin(\gamma/N)}{\gamma/N})^{2} = \mathcal{O} (N\log N)$,$\gamma$is the imaginary part of the zeros of$\zeta (x)$. Why$\sum_{\gamma>0} (\frac{\sin(\gamma/N)}{\gamma/N})^{2} = \mathcal{O} (N\log N)$,$\gamma$is the imaginary part of the zeros of$\zeta (x)$. In Monthomery's Multiplicative Number threory I.... 0 votes 1 answer 92 views ### does a zeta zero ever occur when$Re(\zeta(s))$is at a minimum? The Lehmers' zeros for$\zeta(\frac{1}{2} + it)$are quite close to each other in the value of the ordinate$t$. I am aware of the constraint on the Hardy Z function for which a local positive minimum ... 1 vote 1 answer 106 views ### Proof of Salem's reformulation of Riemann hypothesis. Consider an integral equation: $$\int_{-\infty}^{+\infty}\frac{e^{-\sigma y}f(y)}{e^{e^{x-y}}+1}dy=0$$, where$\sigma\in(\frac{1}{2},1)$In https://arxiv.org/abs/2003.00581 there is written that this ... 1 vote 1 answer 126 views ### How Riemann found his hypothesis? [closed] We know that according Riemann hypothesis all non trivial zeros of dzeta function lie on (0.5, x) line on complex surface. I wonder how Reieman found that idea. Does he just found first few zeros by ... 1 vote 0 answers 106 views ### Gram series in Riemann explicit formula for π(x) Based on Raymond Manzoni's answer https://math.stackexchange.com/a/2822928 we know that Gram series in sum over non-trivial zeros of zeta-function has a slow converging because of large x. And we know ... 3 votes 1 answer 126 views ### Riemann's explicit prime counting formula: how is it piecewise constant? I've heard many times that the distribution of the non-trivial zeros of the Riemann zeta function are hypothesized to match that of the eigenvalues of a random Hermitian matrix (see Wikipedia or this ... 0 votes 0 answers 90 views ### Equal ordinate zeros for the Riemann-zeta function Write$\rho=\beta + i\gamma$and$\rho'=\beta'+i\gamma'$for two distinct non-trivial zeros of the Riemann zeta-function. Is it known that$\gamma\neq\gamma'$? That is, does a proof exist that two ... 1 vote 0 answers 90 views ### Evaluation of$\sum\limits_\Bbb Z Ξ(n)$with the Riemann Xi function on the critical line. This question is inspired from On $$\sum\limits_{x=1}^\infty \text{Ci}(x)$$ for another in a series of an infinite sum of a single function. Another inspiration is the Riemann Xi function$ξ(t)$on ... 0 votes 0 answers 101 views ### Is it any easier to determine if a zero of a polynomial is on the critical circle instead of a Riemann zeta zero on the critical line? The Dirichlet eta function is: $$\eta (s) = \zeta (s) (1-2^{1-s})$$ $$\eta (s) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{s}}$$ Notice that: $$\log(k)=\lim\limits_{n \rightarrow \infty} \frac{\text{... 1 vote 0 answers 85 views ### Can this formula be useful or improved for count prime numbers? Some years ago I made this formula for count prime numbers. It seems more accurate than that created by Gauss but I think it does not make sense and for very large numbers it will not be accurate. ... 1 vote 2 answers 75 views ### How to differentiate the Riemann-Siegel Z-function? Is it well-known/or is there literature for differentiating the Riemann-Siegel Z-function?$$Z(t)=e^{i \theta(t)}\zeta(1/2 +it)$$I have tried differentiating this according to the chain rule but in ... 0 votes 0 answers 51 views ### Riemann zeta function zeros notation In books/papers, we often see the notation that the zeros of the \zeta-function with positive imaginary part are denoted \rho_n in order of increasing height. However, because of the equation \... 1 vote 1 answer 61 views ### Some \liminf for functions with same roots as \zeta(s), in the critical strip The function I have in mind is f(s)=\phi(t)\cdot|\zeta(s)|, where s=\sigma + it, |\cdot| is the modulus, and \frac{1}{2}<\sigma<1 is fixed. Here \phi(t) is a positive, increasing ... 4 votes 1 answer 160 views ### Proof that Riemann Hypothesis is true for first 3,500,000 zeros It was proven by Rosser, Yohe and Schoenfeld that the Riemann Hypothesis is true for the first 3,500,000 zeros and that they are all simple. I’d be interested in reading their paper ‘Rigorous ... 1 vote 0 answers 46 views ### Turing’s Method for verifying Riemann Hypothesis in a range Turing’s method is explained well in p172-175 of the book ‘Riemann’s Zeta Function’ by H.M. Edwards. To understand the finer details, I tried reading the original source of the material in Turing’s ... 0 votes 0 answers 118 views ### Why is that the non-trivial zeros of the Riemann zeta function become ever more frequent with increasing |t|? I understand that, as one climbs the critical line, \sigma = 1/2, with increasing |t|, the non-trivial zeros of the Riemann zeta function \zeta(s), s = \sigma + i \, t, become ever more ... 4 votes 0 answers 128 views ### What's the link between de Bruijn-Newman constant and RH I'm not an expert in analytic number theory, but I've been curious in this last period about it. I stumbled upon de Bruijn-Newman constant during my researches on Wikipedia, where it is written that ... 6 votes 0 answers 242 views ### Why are these integrals equivalent to the Riemann Hypothesis? Riemann Hypothesis is equivalent to the integral equation$$\int_{-\infty}^{\infty} \frac{\log \mid \zeta (1/2+it)\mid }{1+4t^2} \ dt =0$$Many other integral equations exist that are equivalent. How ... 1 vote 0 answers 101 views ### Understanding the ramifications of Riemann Hypothesis Proving Riemann Hypothesis is a million dollar problem, but I am more interested in understanding its ramifications in the practical world. According to many sources, one such effect will be ... 0 votes 0 answers 74 views ### Von Mangoldt function implications$$\psi (x)=x-\sum _{\zeta (\rho )=0}{\frac {x^{\rho }}{\rho }}-\log(2\pi ).$$is the explicit formula for the summatory Mangoldt function. Does this mean x is alway bigger than \psi(x) ? 7 votes 0 answers 132 views ### Bounds for the n^{th}-prime number and the inverse logarithmic integral? Are there any bounds for the n^{th}-prime number p(n) and the inverse logarithmic integral \text{ali}(n)=\text{li}^{-1}(n) under the Riemann Hypothesis?$$|p(n)-\text{ali}(n)|<?$$here is ... 0 votes 1 answer 109 views ### Is the value of the following limit with an infinite number of variables equal to zero or is it not decidable? According to the alternating series test a general term b_n in the sum: \sum\limits_{n=1}^{\infty}(-1)^{n+1}b_n should converge to zero: \lim\limits_{n \rightarrow \infty} b_n = 0 From reuns ... 30 votes 0 answers 721 views ### Colossally abundant numbers and the Riemann hypothesis [This question has lead me to ask a follow up on MathOverflow.] A recent tweet by John Baez has reminded me of the astonishing fact^1 that the Riemann hypothesis (RH) can be disproved by finding a ... 1 vote 0 answers 173 views ### How do you turn an upper bound of the second Chebyshev function into a corresponding lower bound? In Table 5.1 on page 103 of the book The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators by Machiel van Frankenhuijsen the author states:$$\textit{Riemann hypothesis} \iff ... 11 votes 3 answers 472 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 votes 0 answers 87 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)$$$\eta (s)=(1-2^{1-s})\... 2 votes 1 answer 235 views ### Non trivial zeros of Riemann zeta function The non trivial zeros of Riemann zeta function , x\zeta(s) lies in the critical strip 0<\Re(s)<1 Riemann Hypothesis states that all the zeros of Riemann zeta function, \zeta(s) lies on ... 1 vote 0 answers 38 views ### Is there a bound of \Delta(x) related to Riemann prime-counting funtion?$$\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}}},$$where \operatorname{R}(x) is Riemann R function.... 0 votes 1 answer 90 views ### Question Regarding Relationship between the Riemann Zeta Function and the Dirichlet Eta Function So I was looking at the Riemann hypothesis and I saw the relationship between the Riemann zeta function and the Dirichlet eta function which really confused me because I didn't understand how a ... 0 votes 2 answers 90 views ### Pustyl’nikov’s equivalent criteria for RH In this paper (Pustyl’snikov 1999 Russ. Math. Surv. 54 262), Pustyl’nikov proved the following two theorems Theorem 1 All the even derivative of \xi(s) at point s=1/2 are strictly positive. ... 0 votes 1 answer 235 views ### A disproof of RH? [closed] Does this paper of Sondow et. al, in which they propose a disproof of RH, have reasonable arguments ? The Riemann Hypothesis is not true 1 vote 1 answer 124 views ### Does the Mandelbrot Set with a power of -1 produce a Farey Sequence, and if so, can this be connected to the Riemann Hypothesis? I'm by no means an expert in Farey Sequences nor understanding the Riemann Hypothesis - but I do know there is some relationship between them, as alluded to in this question. As such, if the ... 0 votes 1 answer 119 views ### \sum_{\rho}\frac{1}{|\rho|^2}=\sum_\rho\frac{1}{\rho(1-\rho)} Riemann Hypothesis(RH) is equivalent to the statement " https://mathoverflow.net/questions/91280/is-this-sum-of-reciprocals-of-zeta-zeros-correct "$$\sum_{\rho}\frac{1}{|\rho|^2}=\sum_\rho\... 4 votes 1 answer 213 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 votes 0 answers 51 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 votes 1 answer 337 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\... 151 views

The Hardy space $H^p(\mathbf{D})$ is the vector space of holomorphic functions $f$ on the open unit disk that satisfy: $$\sup_{0< r<1}\left(\frac{1}{2\pi} \int_0^{2\pi}\left|f \left (re^{i\... 0 votes 1 answer 61 views ### Prove\sum_{|\alpha|<1, f(\alpha)=0}\log \frac{1}{|\alpha|}= \sum_{\Re(\rho)>1/2}\log\ |\frac{\rho}{1-\rho}| [closed] Let$$ 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\ ... 1 vote
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### 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 ...
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### 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 ...
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### 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?
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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 $I=... -2 votes 1 answer 123 views ### Prove that$\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$... 61 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 :- ... 842 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 ...
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### 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 vote
### 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} =...
### If $z_n$ are the zeros of the zeta function, what is the limit of $\Im{(z_n)}$ as $n$ goes to infinity? 