The Prime Number Theorem is a theorem that describes the distribution of the primes. It says that the number of primes less than or equal to a real number $x$ is asymptotic to $\frac{x}{\ln x}$.

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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< s<...
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Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...
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Does data suggest $| \pi_2 (n) - 2\Pi \int_2^n \frac{dx}{\ln(x)^2} | < \ln(n+2)^2 \sqrt (n+2) $?

Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function. Define $$ t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right| $$ Is it consistent with current ...
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A prime number simplex

Let $\ n\in\mathbf N:= \{1\ 2\ \ldots\}\ $ be a natural number. Let $\ K\ $ be a non-empty finite set of primes. Let $\ \kappa:=|K|.\ $ Consider a $\kappa$-dimensional simplex $\ S_K\subseteq \mathbf ...
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Distribution of primes and near-primes among $\prod p_k \pm 1$

For $n\in \Bbb{Z}^+$ define the statement "$n$ is $k$-social" to mean that $$ \prod_{i=1}^n p_i +1 \mbox{ has exactly } k \mbox{ prime factors} $$ where $p_i$ is the $i$-th prime. So for example $5$ ...
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Density of ratios of an arbitrary increasing sequence of prime numbers

It is well known that the set $\left\{ \frac{p}{q} : p,q \textrm{ prime numbers }\right\}$ is dense in the positive real numbers $\mathbb{R}_{>0}$. Not having a background in number theory, I ask ...
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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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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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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, ...