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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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Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
Sarah's user avatar
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Why could Mertens not prove the prime number theorem?

We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$ where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln ...
Nilotpal Kanti Sinha's user avatar
36 votes
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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 ...
Nilotpal Kanti Sinha's user avatar
32 votes
3 answers
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Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)

I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949). One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$, $$(1) \qquad\qquad \vartheta(x) \...
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Heuristic argument for the prime number theorem?

Here is a bad heuristic argument for the prime number theorem. Let $n$ be a positive integer and assume that PNT holds up to $n$. Then $n$ itself is prime if and only if for each prime $p<n$ the ...
gowers's user avatar
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Is there a Kolmogorov complexity proof of the prime number theorem?

Lance Fortnow uses Kolmorogov complexity to prove an Almost Prime Number Theorem (https://lance.fortnow.com/papers/files/kaikoura.pdf, after theorem $2.1$): the $i$th prime is at most $i(\log i)^2$. ...
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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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Why is the Chebyshev function relevant to the Prime Number Theorem

Why is the Chebyshev function $\theta(x) = \sum_{p\le x}\log p$ useful in the proof of the prime number theorem. Does anyone have a conceptual argument to motivate why looking at $\sum_{p\le x} \log ...
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information-theoretic derivation of the prime number theorem

Motivation: While going through a couple interesting papers on the Physics of the Riemann Hypothesis [1] and the Minimum Description Length Principle [2], a derivation(not a proof) of the Prime Number ...
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Can anything deep be said uniformly about conjectures like Goldbach's?

This is a soft question sparked by my curiosity about the intrinsic depth of Goldbach-like conjectures as perceived by current experts in number theory. The incompleteness theorem implies that, if our ...
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Teaching prime number theorem in a complex analysis class for physicists

This is a question about pedagogy. I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis ...
guest17's user avatar
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Upper bounds for the sum of primes up to $n$

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
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Elementary lower bounds for the number of primes in arithmetic progressions

Some version of the Prime Number Theorem provides the asymptotic behavior of the number of primes in arithmetic progression $qn+a$ with $(q,a)=1$, $n \ge 1$. I was wondering there are Chebyshev-type ...
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14 votes
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A naive question about the prime number theorem

Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$, where $\Lambda(n)$ is the von Mangoldt function. Then as Chebyshev showed, the following equality holds $$\sum_{n\leq x} \psi(x/n)=x\log(x)-x+O(\log(x)).$$ My ...
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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 ...
Daniel Loughran's user avatar
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2 answers
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Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that. $$ p_n = n \...
Nilotpal Kanti Sinha's user avatar
13 votes
1 answer
2k views

Why shouldn't this prove the Prime Number Theorem?

Denote by $\mu$ the Mobius function. It is known that for every integer $k>1$, the number $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^k}$ can be interpreted as the probability that a randomly chosen ...
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12 votes
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Newman's proof of the prime number theorem

I am teaching a graduate course in Complex Analysis and I am covering Newman's proof of the prime number theorem. I have been using the simplified version in the papers of Zagier and Korevaar. However,...
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Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa?

The prime number theorem says on average we can find $\frac n{\log n}$ primes of magnitude $n$. Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ primes. Somehow the fact $e^{\...
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11 votes
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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 -...
Charles Bouillaguet's user avatar
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Squarefree numbers $n$ such that $432n+1$ is also squarefree

This is a second attempt (see Primes $p$ such that $432 p +1$ is prime) Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite? Fact: the number of such ...
user21's user avatar
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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 ...
onionbread's user avatar
11 votes
4 answers
707 views

Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?

Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
H A Helfgott's user avatar
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10 votes
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Quantitative and elementary proofs of the Prime Number Theorem

I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...
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10 votes
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An extremal problem related either to an uncertainty principle on the circle, or else to the prime number theorem

Consider for $X = 1,2, \ldots$ the set $\mathcal{S}_X$ of trigonometric polynomials $f(t) := \sum_{|k| \leq X} c_k e^{2\pi i kt}$ on the circle $\mathbb{T} := \mathbb{R}/\mathbb{Z}$ of degree $\leq X$ ...
Vesselin Dimitrov's user avatar
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The supremum value of $\int f(t) \log{\frac{1}{|t|}} \, dt$ for normalized Fourier pairs non-negative outside of $[-1,1]$

Observe that for any Schwartz function $f \in \mathcal{S}(\mathbb{R})$ having $$ f(0) = \widehat{f}(0) = 1 $$ and $$ f, \widehat{f} \geq 0 \quad \textrm{outside of} \quad [-1,1], $$ the following ...
Vesselin Dimitrov's user avatar
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
10 votes
0 answers
416 views

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 $\...
Tobias Schnieders's user avatar
10 votes
0 answers
739 views

Implications of divergence of $1/\zeta(s) $ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< s&...
Koushik's user avatar
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10 votes
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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 ...
Joël's user avatar
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9 votes
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Strange and non-strange prime numbers, are there infinitely many of them?

Definition. A prime number $p$ is called strange if there exists $k>1$ such that each prime divisior of $p^k-1$ divides $p-1$. Example 3. The prime number $p=3$ is strange as $3^2-1=8$ has the same ...
Taras Banakh's user avatar
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8 votes
2 answers
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Asymptotics of the least common multiple of the first natural numbers

What is $$ \limsup_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \ \ ?$$
Pablo's user avatar
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8 votes
4 answers
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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 ...
pre-kidney's user avatar
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8 votes
3 answers
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Asymptotics for primality of sum of three consecutive primes

We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where $\{p_i\}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc.. The first few values of $R_n$ are: 10, 15, 23, 31, 41, 49, 59, ...
Raj 's user avatar
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8 votes
1 answer
834 views

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 ...
Nilotpal Kanti Sinha's user avatar
8 votes
1 answer
243 views

Main term in the number of sign changes of $\psi(x) - x$

Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$. Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1]. But perhaps that estimate is too crude. ...
ZapMathigan's user avatar
8 votes
1 answer
436 views

Primes in arithmetic progression with a moduli equal to a power of 2

I am currently looking for a result stronger than Siegel-Walfisz theorem, which gives an upper bound on the error term $|\pi(x,a,b)-\frac{\pi(x)}{\phi(a)}|$ for particular $a$. The Siegel Walfisz is ...
Serge Boissot's user avatar
8 votes
1 answer
329 views

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 ...
Marty's user avatar
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8 votes
1 answer
245 views

Asymptotic density of sums of consecutive primes

Call a positive integer respectable if it is a sum of consecutive prime numbers. For example, every prime numbers is respectable. So are $3+5=8$, $2+3+5=10$, $5+7=12$, $3+5+7=15$, $2+3+5+7=17$, $7+11=...
David Corwin's user avatar
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7 votes
3 answers
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a question for the prime counting function

A famous inequality that has been proved by J.B. Rosser and L. Schoenfeld says that $\frac{n}{\ln n-1/2}$ < $\pi(n)$<$\frac{n}{\ln n-3/2} , n\ge 67$. Using this inequality we can prove ...
Konstantinos Gaitanas's user avatar
7 votes
3 answers
607 views

How many primes have the form $(2^p+1)/3$?

Assuming that $p$ is an odd prime. How many primes have the form $(2^p+1)/3$? Is the number finite? Mathematica calculation shows that there are 23 such primes when $p$ ranges over the first 500 ...
Huangjun Zhu's user avatar
7 votes
1 answer
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A curious prime counting approximation or just data overfitting?

I am not sure, if this is a research problem. If not I will move this question to ME: Let $\Omega(n) = \sum_{p|n} v_p(n)$, which we might view as a random variable. Let $E_n = \frac{1}{n} \sum_{k=1}^n\...
user avatar
7 votes
1 answer
3k views

Error term of the Prime Number Theorem and the Riemann Hypothesis

I have read that the Riemann Hypothesis is equivalent to $\pi(x)=\text{Li}(x)+O(\sqrt{x}\log x)$ Is there an analogous statement saying the Riemann Hypothesis is equivalent to $\pi(x)=\frac{x}{\log ...
user16557's user avatar
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7 votes
2 answers
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Legendre's Constant

In a couple of web pages, I see that Legendre's constant is defined to be $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ (for example, here and here). Actually the first uses $\lim_{n \to \infty} (\log(...
user304582's user avatar
7 votes
2 answers
475 views

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) < ...
JoshuaZ's user avatar
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7 votes
2 answers
636 views

How to use the Prime Number Theorem in order to prove Selberg's Formula?

I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory" and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem. This is one of the tasks ...
Juu's user avatar
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7 votes
2 answers
426 views

Divisor sums over values of binary forms of primes

Let $\tau$ be the divisor function, that is $$ \tau(n)=\sharp\{d \in \mathbb{N}, d|n\}. $$ I was wondering if anyone has ever proved an asymptotic estimate for the sum $$S(x):=\sum_{p,q\leq x}\tau(p^...
Dr. Pi's user avatar
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7 votes
0 answers
461 views

On a paper of Alain Connes entitled 'Around Wilson's Theorem '

A relatively recent paper Alain Connes - Around Wilson's theorem introduced the function $$ S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr). $$ In the same paper, he proved that the ...
user avatar
7 votes
0 answers
399 views

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 ...
BDS's user avatar
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6 votes
4 answers
2k views

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 ...
Nilotpal Kanti Sinha's user avatar