Linked Questions
12 questions linked to/from Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
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Is this similarity to the Fourier transform of the von Mangoldt function real? [duplicate]
This question proposed in SEM but no answer and it's interesting to know connection
between Fourier analysis and number theory .
Mathematica knows that the logarithm of $n$ is:
$$\log(n) = \lim\...
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94
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6
answers
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Quasicrystals and the Riemann Hypothesis
Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line:
$$ \zeta(\frac{1}{2} + i k_j) = 0 $$
Let $f$ be the Fourier transform of the sum ...
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3
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What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?
From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...
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Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers
Let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
and let the matrix $T(n,k)$ be:
$$T(n,k)=a(\gcd(n,k)) \tag{2}$$
It has been ...
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Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture
The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambda(m)=\...
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1
answer
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Fourier transform of the von Mangoldt function?
Wikipedia states under the entry for the von Mangoldt function:
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann ...
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The Möbius function as eigenvalues
Let the $N$ by $N$ matrix $A$ be defined by the tetration:
$$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
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Is this irregular curve asymptotic to $\log (2) (\log (x)-\log (2))^2-\frac{\log (2)}{2}$, or is the asymptotic something else?
Let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
And let the matrix $T$ be:
$$T(n,k)=a(\gcd(n,k)) \tag{2}$$
which has the property ...
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What do square roots as minimums have to do with Harmonic numbers?
In an earlier question where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series:
$$\Lambda(m)=\lim_{s\to 1+}\zeta(s)\sum_{d\mid ...
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vote
0
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233
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Primes approximated by eigenvalues?
Let the matrix $T$ be defined by:
$$\displaystyle T(n,k) = -\varphi^{-1}(\operatorname{GCD}(n,k))$$
where $\varphi^{-1}$ is the Dirichlet inverse of the Euler totient function.
$$\varphi^{-1}(n) = \...
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vote
0
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Prove that these linear programming problems are bounded by $O(k^{1/2})$ [closed]
The expanded partial sums of the Möbius inverse of the Harmonic numbers have two out of three properties in common with this set of linear programming problems:
$$\begin{array}{ll} \text{minimize} &...
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Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?
In order to see what happens when taking the functional equation in this form:
$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\...
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