# Questions tagged [prime-number-theorem]

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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### Explicit upper bounds on the number of primes up to the square of the $n^\text{th}$ prime number $p_n$

I'm looking for explicit upper bounds on the number of primes up to the square $m=p_n^2$ of the $n^\text{th}$ prime number. Such estimates can rely on the knowledge of the exact number of primes up to ...
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### What is the intuition behind applying the Mellin Transform to prime distribution?

I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem. I understand that applying the Mellin Transform to the partial sum of the van ...
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### Primality testing by reversible computation using the prime number theorem

Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are ...
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### Geometric prime distribution

Let integers $\ a>1\$ and $\ b\in\mathbb Z\$ be relatively prime (hence $\ b\ne 0).\$ The Dirichlet's prime distribution theorems apply to the arithmetic sequence $$(_aG_b(x) : x\in\mathbb Z)$$...
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### better estimates than the prime number Theorem in Euclidean domains

For a unique factorization domain we know that we have some the analogues of fundamental theorem of arithmetic, and can build elements by using 'building blocks'. For me the easiest examples are ...
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### Averages of Möbius function in arithmetic progressions

It is mentioned in multiple occasions here that the bound $$\mathop{\sum_{n=1}^{N}}_{n\equiv a\mod l} \mu(n) = o(N)$$ is equivalent to the prime number theorem in arithmetic progressions. But I am ...
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### Is $\lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)$?

Let $s>1$ be a real number. We look at the zeta probability function / Zipf probability function defined as: $$P(X = n) = \frac{1}{n^s \zeta(s)}$$ Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is ...
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### Primes in residue classes [duplicate]

For which sets of residue classes are there easy elementary proofs that there are infinitely many primes in them, which don’t require the machinery of proofs of Dirichlet’s theorem? Example: it’s ...
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### Are there highly composite prime gaps?

Definition: Highly composite prime gap The three composite numbers between the consecutive primes $643$ and $647$ each have at least three distinct prime factors. This is the first occurrence of prime ...
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### Asymptotic lower bound for the number of square free with at least two prime factors

In one of Soundararajan's papers, he claims without proof that it is a standard exercise to show that the number $N(X)$ of positive square-free integers $d \equiv 1 \; \bmod \; 8$ less than $X$, with ...
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