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### 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 ...

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### 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< ...

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117 views

### Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.

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132 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 ...

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142 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 ...

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126 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 ...

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105 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, ...

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### Semiprime number theorem with small prime factor

Hardy & Wright, Theorem 437 gives a nice asymptotic for $k$-almost primes less than $x$. Can we say anything if we restrict one of the prime factors of our almost prime to having a small prime ...