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7
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0answers
272 views

Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
6
votes
0answers
269 views

implication of divergence of $1/\zeta(s) $ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< ...
1
vote
0answers
102 views

Prime counting function with a form of finite product using perron's formula

There's a form of complex integral what Riemann obtained to finding $\pi (x)$, $$ \pi^{*}(x)=\int_{L}\frac{\log \zeta (s)}{s}x^{s}ds, (1)$$ we already know that it lead us to the Prime Number ...
1
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0answers
134 views

Limit of Sequence of unusual Prime Product

Let $p_n$ be the nth prime and $p_L$ be closest to its square root: \begin{equation} p_L^2 \approx p_n \approx x \end{equation} Let $\sigma \in Z^+$ be a positive integer constant. Define the ...
0
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0answers
110 views

Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...
0
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0answers
101 views

asymptotics of primes in arithmetic progressions

If $a$ and $q$ are given coprime positive integers, what is the best known error term for $$ \sum_{p<x,\,p\,\text{is prime},\,p\equiv a \pmod q} \frac{\log p}p-\frac{\log x}{\varphi(q)}? $$ Is it, ...
0
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0answers
136 views

Similarities between counting primes and counting integers

Primes can be counted via the explicit logarithmic counting function $\psi(x) \thicksim \pi(x) \log(x)$: $$\displaystyle \psi(x) = x - \log(2\pi) - \frac12 \log(1- \frac{1}{x^2}) - \sum_{\rho} ...