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 $\frac{x}{\ln x}$.
172 questions
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Are 6n ± 1 or more general arithmetic progression forms used as a core tool in advanced Number Theory proofs, beyond basic sieving? [closed]
Is there any research that uses the 6n ± 1 form or the more general form kn ± r to prove more important prime number theorems? (e.g., linked to Dirichlet's theorem on arithmetic progressions)
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Is there a formula to calculate the number of prime numbers between two numbers A and B? [closed]
I’m wondering if there is a known formula (or efficient method) that can calculate the number of prime numbers between any two integers, say A and B.
For example: given A = 10 and B = 30, such a ...
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Can Bellotti’s zero-density bound near the Korobov–Vinogradov region be interpreted as partial evidence toward the Riemann Hypothesis?
In the recent work by Chiara Bellotti, precisely this paper arXiv:2508.02041v1, the author proves that the number of nontrivial zeros of the Riemann zeta function $ \zeta(s) $ with real part $ \beta &...
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Is there a known closed-form expression for the continued fraction of the primes? [closed]
Let $p_n$ denote the $n$-th prime number. I am interested in the constant defined by the simple continued fraction whose partial quotients are the sequence of primes:
$$C = [0; p_1, p_2, p_3, \dots] = ...
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Prime products and powers
NOTATION:
$\quad \mathbb N\ = \ \{1\,\ 2\ \ldots\}$ -- the set of all natural numbers;
$\quad \mathbb P\ =\ \{2\,\ 3\,\ 5\,\ 7\,\ 11\ \ldots\}$ -- the set of all (natural) primes;
$\quad \forall_{x\...
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Formal proofs of the Prime Number Theorem
Which proofs of PNT have been formalized?
As far as I can tell, the situation is the following:
Selberg's elementary proof was formalized in Isabelle in 2007 (Avigad-Donnelly-Gray-Raff) and in ...
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How far can Chebyshev's bounds on $\psi(x)$ go?
Let $\psi(x) = \sum_{n\leq x} \Lambda(n)$, where $\Lambda$ is the von Mangoldt function. For all $x>0$,
$$\frac{\psi(x)}{x} < 1.038821,$$
with the minimum being reached at $x=113$; this has been ...
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Landau-Selberg-Delange (LSD) in arithmetic progressions?
I'm learning about the LSD method. What would be the asymptotic behaviour of the following sums? Is it really LSD what needs to be used here? I am very confused. Is there any reference for tackling ...
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Asymptotics of Riemann's functional equation for the number of primes
In Riemann's famous paper, "On the Number of Prime Numbers less than a Given Quantity" 1 he derives the well known equation:
The equation connecting the logarithm of the Riemann zeta ...
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Conjecture: The decimal expansion of $(p_n)^n$, for $p_n$ the $n$-th odd prime, contains a sequence of consecutive digits whose sum is $p_n$ [closed]
Let $p_n$ denote the $n$-th odd prime number. Consider the number $(p_n)^n$, the $n$-th power of $p_n$.
The Conjecture
The decimal expansion of $(p_n)^n$ always contains a sequence of
consecutive ...
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Do the twin primes occur approximately exponentially often with respect to their position in the twin prime sequence?
I plotted the logarithm of the first $n$ twin primes and noticed that they form an approximately logarithmic curve.
Here is the plot up to 1000 (full scale):
and here is a plot up to 200,000 (full ...
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A Diophantine problem on divisibility involving numbers of the form $p = 4m + 1$
Conjecture. For any $m\in\mathbb{Z}_{\geq 1}$ such that $4m+1$ is a prime number, there exists a pair $ (n,j)\in\mathbb{Z}_{\geq 1}\times\mathbb{Z}_{\geq 0}$, such that
$$(4j+3)⋅n−1∣(4j+3)⋅m+(j+1)$$
...
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Few doubts about "A new elementary proof of the Prime Number Theorem" by Richter
I'm working on Richter's "A new elementary proof of the Prime Number Theorem" paper.
I have some doubt about the proof of proposition 3.1
Here's the reference to the paper: https://arxiv.org/...
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Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter
I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
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What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
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Identities to go from $\sum_{n\leq x} \mu(n) \log \frac{x}{n}$ to $M(x) = \sum_{n\leq x} \mu(n)$?
Let $\mu$ be the Möbius function. Say we have a bound on $\check{M}(x) = \sum_{n\leq x} \mu(n) \log \frac{x}{n}$ of the form $|\check{M}(x)|\leq \epsilon x$ for all $x\geq x_0$.
It is then easy to ...
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First occurrence of formula for $\sum_{n\leq x} \mu(n) \log n$ in terms of $\psi(y)-\lfloor y\rfloor$?
The identity contained in the last two displayed equations in the following passage (from page 110 in Ayoub's An Introduction to the Analytic Theory of Numbers, 1963) gives us right away a simple ...
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Formula for gaps between primes [closed]
The twin prime conjecture refers to:
$$
\liminf_{n\to \infty}\; p_{n+1} - p_{n} = 2.
$$
By reasoning I arrive at the following simple formula for gaps between primes:
\begin{align}
p_{...
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Nontrivial nonrandom properties of prime numbers
What are some nontrivial nonrandom properties of prime numbers. Consider the simple model where each number is prime with probability 1/log(n) by Montgomery and extensions of it. Once you add some ...
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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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Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
user344548
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"Squeezing" the primes?
The logical idea here is to map a curve that encodes the primes into the region $(0,1)^2$ and analyze the distribution there more easily and achieve tight bounds.
To assess the distribution of primes, ...
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Prime number theorem via the explicit formula
Can the prime number theorem be obtained from the explicit formula,
$\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}{\rho}+O(1)$?
Here, $\psi(x)=\sum_{k=1}^\infty\sum_{p^k<x}\log p$
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Generalizations for the PNT to a subset of Dedekind domains?
The classical prime number theorem states that the prime counting function
$$\pi(X) := \# \{ p \leq X \ | \ \text{$p$ prime} \}$$
is asymptotically equal to $X/\log(X)$.
It is also known (and much ...
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Estimating the minimum number of distinct least prime factors found in range of $c$ consecutive integers
When I look at the count of distinct least prime factors for a range of consecutive integers, I am seeing the same minimum number appear again and again. I am wondering if this number represents the ...
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Vinogradov-Korobov prime number theorem for number fields
Without assuming the Riemann hypothesis, the traditional error bound of the prime-counting function $\pi(x)$ is $O(x\exp(-c(\log(x))^{1/2}))$. As shown by the Wikipedia page for the Landau prime ...
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What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?
There is an easy proof of the PNT, just in a few lines, in the book by Julian Havil, "Gamma", pages 201-202. Specifically, Von Mangoldt's formula, which is very easy to derive:
$$
\psi(x) = ...
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Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
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Best possible unconditional partial sum estimate of $\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$:
Consider the following partial sum:
$$S(x,n)=\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$$
Here p runs through primes and $n$ is constant
What is the best possible unconditional( using best known version ...
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Reference for a proof of Euclid's Theorem for the infinitude of primes
I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of ...
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Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
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Density of semiprimes in arithmetic progression
Let $n,a,b$ be integers such that $n$ and $a$ are coprime, and $n$ and $b$ are also coprime. According to the Prime number theorem for arithmetic progressions, the primes which are $a\mod n$ have the ...
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Explicit bounds on number of primes of given size
How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
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Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
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An approach to the prime number theorem with Rademacher variables and a recursive formula for the prime pi function?
Consider the bipartite graphs defined here:
Why is this bipartite graph a partial cube, if it is?
We do random walks on them with equal propability and since the graphs are finite and connected the ...
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Bounded sums involving primes
I'm trying to generalize the Theorem 2.7.1 in [1] where they prove:
$$\sum_{p \leq x} f(p) = \int_{2}^{x} \frac{f(t)}{\log{t}} dt + \epsilon(x)f(x) - \int_{2}^{x} \epsilon(t) f^{'}(t) dt $$
where $\...
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How can I convert Meissel's/Lehmer's formula for prime counting to get sum of primes
Legendre's formula can be very easily be generalised as mentioned here (visible after login) which is like this
${\pi}(v,p)={\pi}(v,p-1)-1.[{\pi}(v/p,p-1)-{\pi}(p-1,p-1)]$
${ \big\downarrow}$
$S(v,p)=...
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How to use prime number theorem In such cases?
Let,
$$A(x)=\sum_{p\leq x}f(p)$$
Where $p$ is a prime number.
Under the Prime Number theorem we have that,
$$\pi(x)=Li(x)+O\left(\frac{x}{e^{a\sqrt{\ln(x)}}}\right) $$
as $x$ approach infinity.
Now,
$$...
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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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Asymptotics of $\operatorname{lcm} ((2-1), (3-1), (5-1), (7-1), (11-1), \dotsc, p_n-1 )$
$\DeclareMathOperator\lcm{lcm}$Let $p_k$ be the $k$th prime number. Set $$L(n) = \lcm(p_1-1, p_2-1, \dotsc, p_n-1). $$
What can we say about the growth of $L(n)$? Trivially, one has that $L(n) < ...
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How essential is the vanishing of the Dirichlet $L$-functions to Dirichlet's theorem on primes in arithmetic progressions?
I seem to recall that the prime number theorem (PNT) is equivalent to the fact that the Riemann zeta function $\zeta(s)$ is non-zero on all of $\text{Re}(s) = 1$ (see https://math.stackexchange.com/...
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Short proof of the error bound in PNT assuming a zero-free strip?
I am looking for a short proof of the fact that $\zeta(z)\neq 0$ for $\Re z>a$ implies the prime number theorem with an error bound $O(x^{a+\varepsilon})$ for any $\varepsilon>0$, which would be ...
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