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10 votes
1 answer
398 views

Vinogradov-Korobov prime number theorem for number fields

Without assuming the Riemann hypothesis, the traditional error bound of the prime-counting function $\pi(x)$ is $O(x\exp(-c(\log(x))^{1/2}))$. As shown by the Wikipedia page for the Landau prime ...
George Bentley's user avatar
-3 votes
1 answer
237 views

L. Gegenbauer's proof of Infinitude of Primes [closed]

I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that L. Gegenbauer proved Infinitude of Primes by ...
math is fun's user avatar
6 votes
1 answer
499 views

Understanding Sylvester' s $1871$ paper of primes in arithmetic progression of the forms $4n+3$ and $6n+5$

The following is the proof of infinitude of primes in arithmetic progression of the form $4n+3$ and $ 6n+5$ done by Sylvester in $1871$ in his paper "On the theorem that an arithmetical progression ...
math is fun's user avatar
5 votes
2 answers
435 views

Proving certain inequality related to Primes

I was reading the following paper. But I can't understand why the last line concerning $\frac{2}{\pi}$ is true. The proof is a work of Sylvester. I would be happy if someone helps me in understanding ...
math is fun's user avatar
4 votes
2 answers
673 views

Counting prime ideals and an explicit Landau prime ideal theorem

Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, and $\mathfrak p$ be a prime ideal of $\mathcal O_K$. Let $x\in \mathbb R^+$, and $N(\mathfrak p)$ be the norm of the prime ideal $\...
var's user avatar
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