Questions tagged [riemann-hypothesis]
Sebastian Wright 
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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 ... riemann-hypothesis divisor-counting-function ramanujan-summation- 163
Questions on summatory function related to non-integer-powers
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 $... number-theory asymptotics riemann-zeta dirichlet-series riemann-hypothesis- 4,581
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.... number-theory prime-numbers riemann-hypothesis- 21
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 ... riemann-zeta riemann-hypothesis- 2,113
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 there is written that this ... reference-request integral-equations riemann-hypothesis- 1,320
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 ... riemann-hypothesis- 129
Gram series in Riemann explicit formula for π(x)
Based on Raymond Manzoni's answer 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 ... prime-numbers riemann-zeta riemann-hypothesis- 11
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 ... prime-numbers riemann-zeta riemann-hypothesis- 309
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 ... complex-analysis number-theory analytic-number-theory riemann-zeta riemann-hypothesis- 301
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 ... sequences-and-series special-functions gamma-function riemann-zeta riemann-hypothesis- 5,322
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: $$\log(k)=\lim\limits_{n \rightarrow \infty} \frac{\text{... polynomials roots riemann-zeta roots-of-unity riemann-hypothesis- 6,790
Can this formula be useful or improved for count prime numbers?
Some years ago I made this formula for count prime numbers. It seems more accurate than that created by Gauss but I think it does not make sense and for very large numbers it will not be accurate. ... number-theory prime-numbers riemann-hypothesis- 65
How to differentiate the Riemann-Siegel $Z$-function?
Is it well-known/or is there literature for differentiating the Riemann-Siegel $Z$-function? $$Z(t)=e^{i \theta(t)}\zeta(1/2 +it)$$ I have tried differentiating this according to the chain rule but in ... calculus complex-analysis riemann-zeta riemann-hypothesis- 1,154
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 $\... riemann-zeta zeta-functions riemann-hypothesis- 1,154
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 ... number-theory riemann-zeta limsup-and-liminf riemann-hypothesis- 3,509
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