# Questions tagged [distribution-of-primes]

Use this tag for questions related to the branch of number theory studying distribution laws of prime numbers among natural numbers.

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### Strange claim about the number of primes between two given numbers.

I recently saw a meme posted to the Facebook group "Mathematical Mathematics Memes." The meme listed some results about the number of primes between two given numbers. The last result listed ...
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### Variations on $\pi(2n)-\pi(n)$

I have taking a glance to the sequence $a_n=\pi(2n)-\pi(n)$ for $n\in\Bbb N$, where $\pi$ is the prime counting function ($\Bbb N$ does not include $0$ in this post). You can see some terms here: ...
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### Distribution of prime gaps - is it an unsolved problem?

Numerical experiments show the distribution of prime gaps conforms to some quite firm constraints. The following plot visualises these constraints - it shows the log of the count of prime gaps against ...
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### Is there a bound of $\Delta(x)$ related to Riemann prime-counting funtion?

$$\Delta (x)=\left(\pi (x)-\operatorname {R} (x)+{\frac {1}{\ln x}}-{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}\right){\frac {\ln x}{\sqrt {x}}},$$ where $\operatorname{R}(x)$ is Riemann R function....
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### Stuck on Derivation of Chebyshev's Estimates for $\pi(n)$

I am trying to develop a simple derivation of Chebyshev's estimates for the prime counting function $\pi(x)$. Stopple's "A Primer of Analytic Number Theory" gives a good start: By ...
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### "Stronger" form of Dirichlet's theorem on primes in arithmetic progressions

Let $a$ and $q$ be two relatively prime positive integers. If I know that $$\lim_{s\rightarrow 1^{+}}\sum_{p\equiv a \bmod q}\frac{1}{p^s}=+\infty,$$ then it is clear that it tells me that there are ...
1 vote
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### Asymptotic of prime number pairs

I am not an expert in number theory, so if someone might know the answer or a lead for the following, it would be greatly appreciated. Let $\mathcal{P}$ be the primes, and let $x$ stand for a natural ...
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### Why is the Theorem of Sylvester and Schur a generalization of Bertrand's postulate?

The Theorem of Sylvester and Schur states that for some n>k, one of n+1,...,n+k-1, has a prime divisor exceeding k. While Bertrand's postulate says that for all positive integers n, there exists a ...
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### Does there exist a a general counting function related to the prime counting function?

Does there exist a a general counting function related to the prime counting function? Say for example I wanted all the positive integer multiples of three less than or equal to N, is there a ... 63 views

### a question about the distribution of primes

Definition : we shall call $p$ a special prime $p_n$, if there is at least one prime of the form $2kp+1$, where $1 \leq k \leq n$ . it is obvious that we can call any Sophie Germain prime, a special ...
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### (soft) Can an efficient closed-form expression for $P_n$ be found? [closed]

I just read several old threads on here with people asking about formulas for primes, and what the implications of having one would be. As everyone was quick to point out, we already have a bunch, in ...
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### A question about the probability of being a prime?

If we chose a random number $a \leq N$, then, the probability for $a$ to be a prime is $\frac{1}{\log N}$. Now, if there are some primes that do not divide $a$, then what is the probability for $a$ ...
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I studied hardy littlewood first conjecture, which predicts the density of primes of special form, so: if I want to know the number of the primes of the form $2kp+1$, where $p$ is prime and $p \leq ... 4 votes 0 answers 79 views ### Do lucky numbers contain arbitrarily long arithmetic progressions? The lucky numbers are defined by a sieve, which results in numbers that asymptotically mirror the prime density$\sim n / \log n$: $$1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, ... 1 vote 1 answer 63 views ### Show that if \lVert a_{pn}\rVert>0 for primes p, then \lVert a_n \rVert\geq0 If we define$$\lVert a_n\rVert = \lim_{N\to\infty}\frac{1}{N}\sum_{k=0}^{N-1}a_k$$To be the mean of a sequence, and we let a_n be a bounded sequence of integers where not only does \lVert a_n\... 1 vote 1 answer 115 views ### A Selberg's sieve application I have a problem concerning Selberg's sieve: I have to prove that, given k>1 a fixed even integer, the number of primes p\leq x such that 1+pk is still a prime is O(\frac{x}{\log^2(x)}). ... 0 votes 0 answers 58 views ### how to predict the density of these prime numbers? if I wanna know how many prime number p less than x, such that : there is at least one prime number of this form 2Kp+1 where K={1,2,3,4,5,6,7,8,9,10}. for example: if x=1000, then the ... 1 vote 1 answer 79 views ### On Bertrand's Postulate By Bertrand's postulate, we know that there exists at least one prime number between n and 2n for any n > 1. In other words, we have$$ \pi(2n) - \pi(n) \geq 1, $$for any n > 1. The ... 1 vote 2 answers 39 views ### Prime reciprocals series proof I am currently struggling with the proof of the prime reciprocals series divergence. I've already proved that :$$\prod_{k=1}^n \frac{1}{1-1/p_k} \longrightarrow +\infty$$Let n be an integer. How ... 1 vote 0 answers 57 views ### Editing digits to find primes Suppose that I am given a random, large (900+ digits) integer. I would like to edit a few of its digits so that the resulting integer is a prime. I am quite flexible on the definition of 'a few'. I ... 0 votes 2 answers 44 views ### M\in\mathbb{N} such that \pi(n)\leq\frac{Mn}{\log n} I know that \pi(n) is approximately \frac{n}{\log n}. Is there any constant M \in \mathbb{N} that satisfies: \forall n \in \mathbb{N}~.~\pi(n) \leq \frac{Mn}{\log n} I need the upper bound ... 1 vote 0 answers 48 views ### Any deducible pattern in this sequence? Is it related to primes? Here's a part of a sequence that I want to understand better: S=\{0,0,1,1,2,2,3,2,4,3,4,3,5,4,5,4,6,5,7,5,8,5,7,5,8,7,...\}. The sequence comes from finding distinct values for f_n(x) where,$$ ... 1 vote 1 answer 899 views ### How to calculate how many primes there are of$N$bits Is there a way to calculate how many primes are exactly$N$bits in length, without generating them? I know that you can calculate how many primes are below$N$, but not how/if you can calculate ... 0 votes 1 answer 74 views ### How to proof the following property? [duplicate] Today I was studying a correctness proof of an algorithm, and in one of the proof steps and did not understand what was written. Be$p$and$n \in N$($N$is the set of naturals numbers), where$p$... 3 votes 0 answers 760 views ### Small gaps between primes Maynard-Tao sieve why it works TLDR: How do we know the probability distribution that Maynard obtains in the small gaps between primes problem is$\mathbb{P}(n+h_{i}) \asymp \frac{\log k}{k}$. I'm interested in why the sieve ... 2 votes 1 answer 84 views ### On the sums of primes It is known that: $$\sum_{k=1}^np_k \sim \frac12n^2\ln{n}$$ The proof of which I cannot find. Here is a related MSE post with an appreciable answer. Now, I wonder how one might go about finding$f(n)...
Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ ...