Tagged Questions
3
votes
2answers
48 views
effective version of Mertens Theorem for the Euler product
I'm referring to the theorem given here, which is
$$\displaystyle\lim_{n\to \infty} \:\: \left(\frac1{\ln(n)} \cdot \left(\displaystyle\prod_{p\leq n} \frac1{1-\frac1p}\right)\right) \;\;\; = \;\;\; ...
0
votes
4answers
74 views
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
1
vote
0answers
43 views
Consequencesof the Hadamard product expression of $L(s, \chi)$
I'm going through my lecture notes and I'm stuck on the proof of
For any $t>0$ and primitive $\chi$ modulo $q$
$$\sum_{\rho=\beta+i \gamma: \Lambda(\rho, ...
5
votes
3answers
90 views
Size of largest prime factor
It is well known and easy to prove that the smallest prime factor of an integer $n$ is at most equal to $\sqrt n$. What can be said about the largest prime factor of $n$, denoted by $P_1(n)$? In ...
4
votes
2answers
87 views
Using sum of logarithms of primes to prove the number of primes up to $n$ is $O(n/\log n)$
I need to show that the number of primes up to $n$ (i.e. $\pi(n)$) is $O(n/\log n)$.
In the previous exercise of this question I proved that ${\displaystyle \sum_{i=1}^{\pi(n)}\log p_{i}} \leq Cn$ for ...
4
votes
1answer
117 views
Is this the way to estimate the amount of lucky twins?
To estimate the amount of prime twins between $3$ and $x$ we just take $x \prod_{p}(1-2/p)$ where $p$ runs over the primes between $3$ and $\sqrt x$. Lucky numbers are similar to prime numbers. Does ...
2
votes
1answer
50 views
What error bound would an epsilon closer to the Riemann hypothesis give?
$s=1$ line gives: $$\psi(x) = x(1+o(1))$$
classical zero free region gives: $$\psi(x) = x + O(x e^{-c \sqrt{\log x}})$$ for some positive constant $\delta$
RH gives: $$\psi(x) = x + ...
0
votes
0answers
89 views
Divergence of the Derivative of the Prime Counting Function
On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written
$$
\pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$
with $ \operatorname{R}(z) = ...
0
votes
1answer
147 views
Evaluating the sum of $\omega(n)$ in an arithmetic progression [closed]
Let $\omega(k)$ count how many distinct prime factors k has,
Then I can prove that for any coprime integers $a,b$
$$\lim_{n\to\infty}\frac{\sum_{k=2}^n\omega(ak+b)}{\sum_{k=2}^n\omega(k)}=1$$
Does ...
3
votes
1answer
244 views
Why the Riemann hypothesis doesn't imply Goldbach?
I'm interested in number theory, and everyone seems to be saying that "It's all about the Riemann hypothesis (RH)". I started to agree with this, but my question is:
Why then doesn't RH imply the ...
2
votes
1answer
117 views
Vonmangoldt sums
The dirichlet series for the Vonmangoldt function, $\Lambda(n)$, which is equal to zero when $n$ is not a prime a power, and $ln(p)$ when it is a prime power say, $n=p^j$, is
...
6
votes
1answer
170 views
Prime number sum
Let $p$ denote a prime, and let $\{x\}$ denote the fractional part of $x$.
Suppose that the following statement is true for all non-integer real numbers $x$:
$$\lim_{n\to\infty}\frac{\sum_{p\leq n}^\ ...
1
vote
3answers
65 views
Questions about assigning a probability to a randomly chosen large integer $n$ being prime
I heard this question a few days ago, so reciting from memory:
If I were to randomly choose an arbitrarily large positive integer $n$, could I write a function that determines the likelihood of it ...
0
votes
1answer
51 views
Polynomial that permutes residue classes
Prove that for any integers $d, e > 1$, the polynomial $f$ with integer coefficients permutes the residue classes modulo $p^d$ if and only if it permutes the residue classes modulo $p^e$ where $p$ ...
2
votes
1answer
65 views
$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $
Let $\ π(x)$ denote the prime counting function , i.e. the number of primes not exceeding $x$
Then does $$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $$ exist for all real $δ$ $∈ ( 0 , 1 )$
2
votes
2answers
213 views
Is there a way to show that $\sqrt{p_{n}} < n$?
Is there a way to show that $\sqrt{p_{n}} < n$?
In this article, I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ ...
2
votes
1answer
143 views
Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?
Let $x$ be a positive real number.
Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$.
Call this function $f(n,x)$.
Can we give good upper and lower bounds of $f(n,x)$ ...
6
votes
1answer
192 views
how to prove this extended prime number theorem?
A Generalized Prime Number Theorem?
Conjecture
Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest ...
2
votes
1answer
253 views
Chebyshev's first $\vartheta(x)$ function question
This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
2
votes
0answers
108 views
Partial summation of a harmonic prime square series (Prime zeta functions)
I am trying to find the following series:
$S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$
...
1
vote
1answer
130 views
Two Representations of $\log \zeta$
I was looking for representations of $\log \zeta$ and found these two:
$ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted ...
7
votes
2answers
170 views
Generalized PNT in limit as numbers get large
If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is:
$$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
2
votes
1answer
131 views
A prime conjecture
Let $n_k$ for $k=1,2,...,i$ be a finite sequence of positive integers, with $i>1$ and $n_1=0$. If there is a prime p such that for every positive integer m, one or more integers in {${(m+n_k)|1\leq ...
3
votes
1answer
83 views
Exclusive prime factors
Let $S$ be an finite or infinite subset of the primes. Let $f(x)=1$ if $x$ has no factors in $S$. If not, $f(x)=0$. Is there a way to calculate the limit $\displaystyle\sum_{n=1}^{x} f(n)/x$, as $x$ ...
6
votes
1answer
99 views
Limit of ratios of numbers with $m$ factors and primes
This is my first question.
Let $a_1, a_2,\ldots, a_k$ be natural numbers $\leq n$ with $m$ prime factors.
Let $p_1, p_2, \ldots, p_r$ be the prime numbers $\leq n$.
Let $$C_{m,n} = ...
3
votes
1answer
217 views
Is there a simple way to prove Bertrand's postulate from the prime number theorem?
Is there a simple way to prove Bertrand's postulate from the prime number theorem?
The prime number theorem immediately implies Bertrand's postulate for sufficiently large $n$, but it fails to ...
5
votes
0answers
106 views
Limit inferior of the quotient of two consecutive primes
I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a ...
5
votes
1answer
187 views
Always a prime between $x$ and $x+cf(x)$
What is the asymptotically slowest growing function $f(x)$, such that there exists constants $a$ and $b$, such that for all $x>a$, there is always a prime between $x$ and $x+bf(x)$?
$f(x)=x$ ...
6
votes
2answers
202 views
Question regarding Von-Mangoldt function.
Let $\psi(x) := \sum_{n\leq x} \Lambda(n)$ where $\Lambda(n)$ is the Von-Mangoldt function.
I want to show that if $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =1 $$ then also $$\lim_{x\rightarrow ...
5
votes
0answers
106 views
Density of products of a certain set of primes
I have an infinite set S of prime numbers with relative density 0 (that is, $\lim_ns_n/p_n=\infty$ with $S=\{s_1,s_2,\ldots\}$ and $s_1 < s_2< \cdots$). I would like to find the size (in some ...
8
votes
1answer
163 views
Rate of convergence of series of squared prime reciprocals
It is well known that $\sum_{p \text{ prime}} \frac{1}{p}$ diverges, and in fact - it behaves like log of the harmonic series:
$$
\sum_{p \le x} \frac{1}{p} = \log \log x + O(1).
$$
It is also well ...
5
votes
1answer
131 views
Inequality appearing in proof of Mills' Theorem
I'm reading this (very short, 1 page long) paper by W.H. Mills where he determines that there exists a real number $A$ such that $f(n) = \lfloor {A^3}^n \rfloor$ is a prime number for all positive ...
3
votes
2answers
444 views
Erdős and the limiting ratio of consecutive prime numbers
The following is a piece of math lore from the late forties, which was described in an Intelligencer article entitled "The Elementary Proof of the Prime Number Theorem". It reads:
Turán, who was ...
12
votes
1answer
245 views
Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$
How do I prove the following:
$$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$
I tried induction, but I didn't know how to go on because I don't have a look at all numbers.
...
3
votes
2answers
444 views
Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?
This is follow up question on this: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \mathrm{Re}(s) < 1 $?
There it is stated that:
$$
\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + ...
6
votes
2answers
600 views
Sum of reciprocal prime numbers
How can the following equation be proven?
$$ \forall n > 2 : \sum_{p \le n}{\frac1{p}} = C + \ln\ln n + O\left(\frac1{\ln n}\right), $$ where $p$ is a prime number.
It's not homework; I just ...
0
votes
1answer
71 views
Compare two sums using prime number theorem
Let $$A(x)=x\sum_{p \leq x} 1, B(x)=\frac{3}{5}\sum_{x<p\leq 2x}p$$ Using prime number theorem, we have $A(x)\sim\frac{x^2}{\log{x}}$, but how to obtain an estimation for $B(x)$?
6
votes
3answers
150 views
Showing $\pi(ax)/\pi(bx) \sim a/b$ as $x \to \infty$
I'm having a bit of a problem with exercise 4.12 in Apostol's "Introduction to Analytic Number Theory". I don't think it's supposed to be a very hard exercise, it's the first one in its section ...
5
votes
3answers
242 views
Evaluate $d(n!)$
An exercise:
Using the prime number theorem find an asymptotic expression for $d(n!)$ where $d$ is the number of divisors.
9
votes
1answer
318 views
Polar Density of a Set of Primes
In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes of a number field $K$ as follows:
Definition: If some $n$th power of the function
$$\zeta_{K,A}(s) = ...
2
votes
2answers
106 views
Evaluating two limits related to prime numbers
How to find these limits
$\displaystyle\lim_{n\to\infty}\left(\ln(\ln(n)) - \sum_{k=2}^n\frac1{k \ln(k)}\right)$ ?
and $\displaystyle\lim_{n\to\infty}\left( \ln(\ln(n)) - ...
1
vote
0answers
117 views
Binary sequences in primes
Is anything known about these problems?
If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...
12
votes
1answer
429 views
What is the probability that some number of the form $10223\cdot 2^n+1$ is prime?
I (David Speyer) took the liberty of doing a fairly major rewrite of this question. I hope I haven't gone too far, but I think there is an interesting question hiding here.
Sierpinski proved that ...
5
votes
3answers
142 views
For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?
The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges.
What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
6
votes
2answers
367 views
Equivalence to the prime number theorem
I was just reading this question and answer: How will this equation imply PNT
and it raised a whole new question:
Given that $\sum_{n\le x} \Lambda(n)=x+o(x)$, prove that $$\sum_{n\le x} ...
29
votes
2answers
999 views
Small primes attract large primes
$$
\begin{align}
1100 & = 2\times2\times5\times5\times11 \\
1101 & =3\times 367 \\
1102 & =2\times19\times29 \\
1103 & =1103 \\
1104 & = 2\times2\times2\times2\times ...
6
votes
0answers
225 views
Question about a proof in Iwaniec-Kowalski's Analytic Number Theory book
My question is about the end of the proof of theorem 1.1, in page 27.
Namely, it is stated that whenever we have a multiplicative function $f:\mathbb{N} \to \mathbb{C},$ let the sequence ...
14
votes
2answers
420 views
Primes sum ratio
Let
$$G(n)=\begin{cases}1 &\text{if }n \text{ is a prime }\equiv 3\bmod17\\0&\text{otherwise}\end{cases}$$
And let
$$P(n)=\begin{cases}1 &\text{if }n \text{ is a prime ...
4
votes
2answers
165 views
How to prove this inequality using prime number theorem
Define $s_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime number, now how to show that
$$\lim_{n \rightarrow \infty} \inf \frac{s_n}{\log n} \leq 1$$
I used the result from the prime number theorem: ...
3
votes
3answers
192 views
Convergence of prime series
Where can I read about convergence of series constituted of prime number such as the following:
$$\sum_p \frac{1}{p (\log{p})^\alpha}\;?$$ How does convergence depend on $\alpha$?
