# Tagged Questions

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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### Prime Ideal Theorem for Real Quadratic Number Rings over Hyperbolic sectors?

As a follow-up to my previous question about variants of Prime Ideal Theorems about Imaginary quadratic number rings found here, I am now asking about the existence of such a companion result for Real ...
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### A variant of the equidistribution of primes in an imaginary quadratic number ring

It is known that the arguments of prime elements of $\mathbb{Z}[i]$ are equidistributed in $(0,2π)$ (by Theorem 5.36 of Iwaniec and Kowalski, or one of Kubilius' papers cited below). This theorem ...
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### Lexicographic distribution of irreducible polynomials

Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its ...
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### How to count fixed-sized subsets of pairwise co-prime numbers less than a prime, satisfying an additional ‎constraint‎?

In part of my research, I need to count (or find a polynomial bound for) the number of ‎possible ‎ways to select $n$ distinct integers less than the prime $p$, say $r_1, r_2, …, r_n$, which ‎are ...
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### Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)

What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)? In other words, I am curious about the state of the ...
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### Given an even integer N, what is the minimum set of primes such that any even number x <= N can be expressed as the sum of two primes from the set?

Given an even integer N, what is the minimum set of primes such that any even number $x \leq N$ can be expressed as the sum of two primes in the set? Goldbach's conjecture said Every even integer ...
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### Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...
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### The prime number $2$ [duplicate]

Possible Duplicate: Why is 2 so odd? I have read few books and articles, almost all of them refer that any prime $p>2$. Just wondering why it has to be $>2$?
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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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### Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
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### Probability that randomly chosen integers from a restricted set of natural numbers are coprime

We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is $$P(k) = \frac{1}{\zeta(k)}.$$ I am looking at a special case of ...
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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: $$p_L^2 \approx p_n \approx x$$ Let $\sigma \in Z^+$ be a positive integer constant. Define the ...
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### Name of a conjecture on difference of prime numbers? [closed]

Hello Dear there is a conjecture for which I do not know how it is called. The conjecture is: Every even number can be always written as the difference between two prime numbers. Could you ...