All Questions
84 questions
5
votes
1
answer
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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 ...
- 53
3
votes
0
answers
191
views
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}...
- 20.2k
-3
votes
1
answer
201
views
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_{...
- 13
0
votes
0
answers
122
views
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 ...
- 727
5
votes
1
answer
736
views
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
2
votes
2
answers
424
views
"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, ...
- 383
11
votes
2
answers
1k
views
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":
$$
\sum_{p \leq n} \frac{\log p}{p -...
4
votes
1
answer
601
views
Reference for a proof of Euclid's Theorem for the infinitude of primes
I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of ...
- 17.9k
13
votes
4
answers
2k
views
Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
- 21.1k
4
votes
1
answer
251
views
Density of semiprimes in arithmetic progression
Let $n,a,b$ be integers such that $n$ and $a$ are coprime, and $n$ and $b$ are also coprime. According to the Prime number theorem for arithmetic progressions, the primes which are $a\mod n$ have the ...
- 654
2
votes
1
answer
283
views
Explicit bounds on number of primes of given size
How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
- 456
10
votes
0
answers
416
views
Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
0
votes
0
answers
136
views
Bounded sums involving primes
I'm trying to generalize the Theorem 2.7.1 in [1] where they prove:
$$\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 $$
where $\...
- 109
0
votes
2
answers
302
views
How can I convert Meissel's/Lehmer's formula for prime counting to get sum of primes
Legendre's formula can be very easily be generalised as mentioned here (visible after login) which is like this
${\pi}(v,p)={\pi}(v,p-1)-1.[{\pi}(v/p,p-1)-{\pi}(p-1,p-1)]$
${ \big\downarrow}$
$S(v,p)=...
- 103
0
votes
1
answer
195
views
Geometric prime distribution
Let integers $\ a>1\ $ and $\ b\in\mathbb Z\ $ be relatively prime (hence $\ b\ne 0).\ $ The Dirichlet's prime distribution theorems apply to the arithmetic sequence
$$ (_aG_b(x) : x\in\mathbb Z) $$...
- 4,776
0
votes
0
answers
144
views
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 ...
- 1,561
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 ...
- 5,460
2
votes
1
answer
460
views
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/...
- 781
20
votes
2
answers
4k
views
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 ...
- 3,871
1
vote
1
answer
186
views
Comparing densities of different gapped primes (twin, cousin, sexy...) [closed]
In this experiment, I have checked how many times different gapped primes occur out of the first 10000, 100000, 1000000 first primes.
Please view the following as ($X$:$Y$) where $X$ represents the ...
- 123
2
votes
0
answers
312
views
Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem
Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$?
If so: Let $s_{0}$ ...
- 129
4
votes
1
answer
395
views
Mertens formulas aren't enough for prime number theorem
For the primes it's true that
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x)
$$
where, $M$ is suitable constant, and, moreover, the prime number theorem gives that
$$
\lim_{x\to\infty}\frac{\...
- 41
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 ...
- 129
-2
votes
1
answer
181
views
Polynomials of minimum degree that interpolate primes in intervals
Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...
- 1,826
9
votes
1
answer
699
views
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 ...
- 41.8k
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 ...
- 233
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 ...
- 233
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=...
- 15.4k
5
votes
0
answers
137
views
Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?
This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO.
For every ...
- 5,460
29
votes
2
answers
3k
views
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 $i(\log i)^2$. ...
- 13.9k
0
votes
1
answer
474
views
An upper bound for $\sqrt{p_{n+1}}$
Let $C$ be a positive constant. Is it true that for all sufficiently large integers $n$ the inequality $$\prod_{i=1}^n (1+\frac{1}{\sqrt{p_i}})>C\sqrt{p_{n+1}}$$ holds? (Here with $p_k$ is denoted ...
0
votes
1
answer
256
views
Lower bound for $\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)}$
What is the best lower bound known for $$\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)},$$ where the product is taken over all the primes(congruent to $3$ modulo $4$) less than or equal to $n$.
3
votes
2
answers
386
views
Prime divisors of $\prod_{i=1}^n (i^2+1)$
Is it true that for every positive integer $n$ there is a prime $p>n,$ which divides $\prod_{i=1}^n (i^2+1)$ ?
5
votes
0
answers
193
views
Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
7
votes
1
answer
1k
views
A curious prime counting approximation or just data overfitting?
I am not sure, if this is a research problem. If not I will move this question to ME:
Let $\Omega(n) = \sum_{p|n} v_p(n)$, which we might view as a random variable.
Let $E_n = \frac{1}{n} \sum_{k=1}^n\...
user6671
13
votes
1
answer
2k
views
Why shouldn't this prove the Prime Number Theorem?
Denote by $\mu$ the Mobius function. It is known that for every integer $k>1$, the number $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^k}$ can be interpreted as the probability that a randomly chosen ...
- 1,019
20
votes
4
answers
2k
views
Can anything deep be said uniformly about conjectures like Goldbach's?
This is a soft question sparked by my curiosity about the intrinsic depth of Goldbach-like conjectures as perceived by current experts in number theory. The incompleteness theorem implies that, if our ...
- 2,912
6
votes
2
answers
411
views
A simultaneous generalization of the Grunwald-Wang and Dirichlet Theorems on primes
By Grunwald-Wang Theorem, if for some odd number $n$ the equation $x^n=a$ has no solutions in $\mathbb Z$, then the equation $x^n=a\mod p$ has no solutions for some prime number $p$. I am interested ...
- 41.8k
3
votes
0
answers
206
views
Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
- 11.3k
2
votes
2
answers
1k
views
What is the natural density of hyper prime numbers?
What do we mean by hyper prime numbers? Well, roughly speaking they are natural numbers which are prime with respect to hyperoperators in arithmetic such as exponentiation, tetration, pentation, et ...
3
votes
2
answers
2k
views
Prime counting. Meissel, Lehmer: is there a general formula?
I am looking for a general forumla to count prime numbers on which the Meissel and Lehmer formula are based:
$$\pi(x)=\phi(x,a)+a-1-\sum\limits_{k=2}^{\lfloor log_{p_{(a+1)}}(x) \rfloor}{P_k(x,a)}$$
...
- 91
4
votes
0
answers
672
views
Euclides' sieve
This is probably a well-known problem. Given a set or multiset of natural numbers let us construct its "Euclides" closure: on each step we take all possible products $P_i$ of the elements in the set, ...
- 5,043
2
votes
2
answers
1k
views
Estimates for Sum of Prime Factors and Number of Prime Factors
Given a positive integer $n$, I've workout out a formula which involves the expression "sum of distinct primes dividing n" minus "number of distinct prime factors of n."
Are there any known ...
- 329
7
votes
2
answers
2k
views
Legendre's Constant
In a couple of web pages, I see that Legendre's constant is defined to be $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ (for example, here and here).
Actually the first uses $\lim_{n \to \infty} (\log(...
- 595
3
votes
1
answer
230
views
Double max of a particular sum in Montgomery-Vaughan
In the Montgomery-Vaughan's paper ''The exceptional set in Goldbach's problem'',
they estimate the following sum:
$$\displaystyle \max_{0<y\leq x}\max_{0<h\leq x} \left(h+\frac{x}{P}\right)^{-1}...
3
votes
1
answer
224
views
PNT analog for primes inside a structured set
Let $\Bbb T$ be the set of all square free integers with ordering derived from $\Bbb N$. Essentially $PNT$ says if you pick $\log N$ integers less than $N$ you can expect one of them to be prime.
...
- 13.9k
4
votes
0
answers
318
views
Prime powers between $x$ and $x+x^\theta$
By the result of Baker, Harman, Pintz (http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf), for any sufficiently large $x$ the interval $[x-x^{21/40},x]$ contains a prime number. This ...
- 41.8k
1
vote
1
answer
317
views
An explicit value for a bound proof
I saw a proof that $|p_n - li^{-1}(n)| \leq n e^{-c \sqrt{\ln(n)}} $,
without saying anything about $c$ !
My questions is, what the explicit value of $c$ ??
It just says for some number $c$ without ...
user95470
1
vote
1
answer
867
views
$n$th prime: a better approximation
Let $p_n$ be the $n$-th prime, then from Wikipedia I got that
$p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.
What is a ...
user95470
32
votes
3
answers
8k
views
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) \...
- 587