0
votes
0answers
20 views

Upper bound number of distinct prime factors

I want to prove that if $\omega (n)$ is the number of distinct prime factors of $n$ for $n>2$ we have $\omega (n) \leq \frac{\ln n}{\ln \ln n} + O(\frac{\ln n}{(\ln \ln n)^2})$. How can I do this? ...
0
votes
0answers
15 views

How to show that $\sum_p \int_{p^m}^\infty f(x) dx = \int_0^\infty \pi(x^{1/m}) f(x) dx$

How do you show that for some function $f(x)$, $$\sum_p \int_{p^m}^\infty f(x) dx = \int_0^\infty \pi(x^{1/m}) f(x) dx$$ where the sum on left is taken over the set of all prime numbers $p$ and ...
0
votes
0answers
15 views

Upper bound for number of primes in an interval

Let $S(x,y)$ be the number of primes $p$ in $(x, x + y]$ such that also $p + 6$ and $p + 12$ are primes. I know that $$ T(x, y) \leq 48 c \frac{y}{\log^3 y} \left( 1 + O \left ( \frac{\log \log ...
0
votes
0answers
18 views

Summation of Legendre symbol

Let $\chi_{2,q}$ be the real Dirichlet character modulo a prime $q>2$, which is not the principal one (the so-called Legendre symbol). Is it true that $$ \sum_{n=0}^{+\infty} ...
2
votes
2answers
25 views

Summation of non-principal real Dirichlet character

Let $q > 3$ be a prime and $$ S_q := \sum_{k=1}^{q-1} \chi_{2,q} (k) \, k, $$ where $\chi_{2,q}$ is the real Dirichlet character modulo $q$ which is not the principal one. I have to prove that ...
1
vote
3answers
38 views

Divisor function convolution

I need some help to prove that $$ (d*d)(p^k) = \frac{(k+3)(k+2)(k+1)}{6} \qquad \forall p \in \mathcal{P},\quad \forall k \in \mathbb{N}, $$ where $d$ is the divisor function and $\mathcal{P}$ the set ...
2
votes
1answer
109 views

Between $n$ and $2n$ there is always a prime number. [duplicate]

Between $n$ and $2n$ there is always a prime number. I was thinking of this and looked it up on the google to find that this is true. Now, I am wondering what is the proof for it? Does any ...
4
votes
1answer
78 views

What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
2
votes
1answer
40 views

Integers Free of Small Prime Factors

I am trying to understand (a version of) the elementary proof of the Prime Number Theorem. I've been following Tenenbaum and Mendès France's book The Prime Numbers and Their Distributions. My goal is ...
0
votes
2answers
37 views

Asymptotic behaviour of $\prod_{p \leq x} (1 + 4/(3p) + C p^{-3/2})$

I'm reading Montgomery and Vaughan and in it they state quite simply \begin{equation} \prod_{p \leq x} \left(1 + \frac{4}{3p} + \frac{C}{p^{3/2}} \right) \ll (\log x)^{4/3} \end{equation} as $x ...
0
votes
0answers
44 views

Number of prime factors of Mersenne numbers

Let $p$ be a prime and let $M_p = 2^p-1$. Is it known whether the number of prime factors of $M_p$ is unbounded above as $p \to \infty$? Also do the probabilities estimating the chance that $M_p$ is ...
5
votes
1answer
60 views

Is any elementary proof important (beside Selberg's work) ?

Is any elementary proof important (beside Selberg's work) ? Plus, why is the elementary proof of prime number theory by Selberg so important ? Selberg was awarded the Field medal is mainly because ...
0
votes
1answer
95 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
2
votes
3answers
56 views

Smallest prime factor of a Mersenne number

The Mersenne numbers $M_n$ are integers of the form $2^n-1$, where $n$ is a positive integer. In the case when $n$ is a prime, are there any results known on the smallest prime factor, $p_n$, of ...
2
votes
1answer
42 views

A slightly various form of Dirichlet's theorem on arithmetic progressions

Are there infinitely many primes of the form $2n(n+1)+1$?
0
votes
0answers
102 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
1
vote
0answers
52 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
1
vote
1answer
37 views

Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} ...
5
votes
2answers
194 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
1
vote
1answer
71 views

Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. ...
1
vote
1answer
54 views

On Newman/Zagier's proof of PNT

I have just got this paper: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf and I have a serious doubt: When proving that soft Tauberian theorem he explicitly uses ...
1
vote
1answer
75 views

Prove or disprove that $\forall k\in\mathbb N$ there exist tree consecutive primes such that $p_i-p_{i-1}\gt k$ and $p_{i+1}-p_{i}\gt k$

Prove or disprove that for every positive integer $k$, there exist tree consecutive primes $p_{i-1}, p_i, p_{i+1}$ such that $p_i-p_{i-1}\gt k$ and $p_{i+1}-p_{i}\gt k$. It's well known that ...
3
votes
2answers
55 views

$\pi(x)\leq \frac x{f(x)}$ for some unbounded function $f(x)$

Let $\pi(x)$ denote the number of primes $\le x$. Can one prove $$\pi(x)\leq \frac x{f(x)}$$ for some function $f(x)(x\gt0)$, and $f(x)$ is unbounded? Please do not refer to prime number ...
2
votes
1answer
174 views

How do we prove $p_n\sim n\log(n\log(n))$ from the Prime Number Theorem?

Let $p_n$ be the $n$th prime. Could someone please help me with the steps between $\pi(n)\sim\dfrac{n}{\log(n)}$ and $n=\pi(p_n)$, to the statement $p_n\sim n\log(n\log(n))$?
4
votes
1answer
97 views

Is this sequence monotonically decreasing?

Let $a_n = \frac{p_n - p_{n-1}}{p_n \log p_n}$ where $p_n$ denotes the $n$-th prime. Is this sequence decreasing (or decreasing after some $N$)?
2
votes
1answer
42 views

In how many ways can a number be factorized over the field $\mathbb{Z}_p$ into two numbers?

For example, over the field $\mathbb{Z}_5$, we can factor number 4 into two numbers in three different ways, i.e. 4=4$\times$1, 4=2$\times$2, and 4=3$\times$3. I am looking for a general formula of ...
3
votes
0answers
76 views

Major arcs in the proof that every odd number is the sum of at most 5 primes

In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
4
votes
1answer
46 views

Motivation for using $L(1,\chi)$ in the proof of Dirichlet's Theorem

Having read the proof of Dirichlet's Theorem on the infinitude of primes in arithmetic progressions, I am left wondering what his motivation for studying $L(1,\chi)$ was and why it is reasonable that ...
4
votes
2answers
70 views

Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
2
votes
1answer
73 views

Error term of the prime number theorem in arithmetic progressions

It is known that if $(a, q)$ and $q\le (\ln x)^N$, then the following is true $$\sum_{k\le x, k\equiv a\pmod{q}}\Lambda(k) = \frac{x}{\phi(q)} + O(x\exp(-C\sqrt{\ln x}))$$ where $C$ depends only on ...
1
vote
1answer
76 views

Prime Counting: Relationship between Chebyshev's function and the Prime counting function

How do I show that if $\psi(x)=x+O(x^{1/2}\log^2(x))$ then $\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$ Where $\psi(x)$ is Chebyshev's second function and $\pi(x)$ is the prime counting ...
2
votes
1answer
36 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
0
votes
1answer
36 views

Proof of convergence of $L'\left(1,\chi\right)$

can someone give me a good reference for a clear proof of the convergence of $L'\left(1,\chi\right)$, $\chi$ real-valued, non-principal Dirichlet character? Thanks in advance.
0
votes
1answer
47 views

Questions about the proof that every odd integer is the sum of 5 primes

In http://arxiv.org/pdf/1201.6656.pdf, Tao proved that all odd numbers greater than 1 are the sum of 1, 3, or 5 primes. In page 36-37, he uses the fact that for all $x > 1.1\times10^{10}$, there ...
5
votes
1answer
60 views

Sum of a certain series related to the primes

It is well known that $$\sum_{n > 0}\frac{1}{n}$$ diverges, but $$\sum_{n > 0}\frac{1}{n^2} = \frac{\pi^2}{6}$$ converges. Similarly, $$\sum_{p}\frac{1}{p}$$ diverges, but $$\sum_{p} ...
2
votes
1answer
43 views

Second part of Eloi's Conjecture

We know that "There exist some real k such that ∀ integer n>1 the integer part of k∗nln(n) is always prime?" is false (prove here Is there a $k$ for which $k\cdot n\ln n$ takes only prime values? ) ...
1
vote
0answers
57 views

Integration by parts of the Logarithmic Integral

I am doing work on analytic number theory, and I am currently looking at the Prime Number Theorem, that is $$\pi(x) \sim Li(x)$$ Some of my sources say that I can do integration by parts on the ...
2
votes
1answer
121 views

How to prove the convergence of a series of prime numbers

I have a bit of a problem proving that the series: $$ \sum_{p\leq x} \frac{p\ln\left(p\right)}{x^2} $$ where the sum is extended over all prime numbers, converges to 0.5. Any ideas? Thanks in ...
5
votes
0answers
109 views

Are $ut + 1$ and $ut + t + 1$ both prime for some t for any $u$?

Conjecture : For any natural number $u$, there is a natural number $t$ such that $ut + 1$ and $ut + t + 1$ are both prime. So we get a solution of the equation $$au - b(u+1) = -1$$ with prime ...
1
vote
2answers
77 views

Function approximating this product

Is there any function approximating, for large values of $p$, the quotient between the product of all primes and the product of all primes $-1$? Basically: $2/1 \cdot 3/2 \cdot 5/4 \cdot 7/6 \cdot ...
7
votes
1answer
155 views

Is the Green-Tao theorem a consequence of the Euler's theorem?

The Erdős-Turán conjecture states that If $A\subset\mathbb{N}$ is such that $$ \sum_{n\in A} \frac{1}{n} = \infty,$$ then $A$ contains arithmetic progressions of any given length. I'm ...
2
votes
0answers
69 views

Liouville function and PNT

The Big Omega function is defined as the number on non-distinct prime factors of an integer. I.e. $\Omega (2^a3^b...p^z)=a+b+...+z$, and the Liouville function is defined as ...
5
votes
1answer
117 views

On a constant defined by Ramanujan.

In the second letter to Hardy Ramanujan writes about the number of prime numbers less than $n$ there he writes. Here this constant $\mu$ facinated me . What is its closed form? and How to compute ...
2
votes
2answers
91 views

Prime number theorem and how many primes are close to $x$ for sufficiently large $n$

The prime number theorem states: $$ \lim_{x-> \infty}{\frac{\pi(x)}{\frac{x}{ln(x)}}} = 1 $$ I was trying to get a better understanding on the intuition on that statement and more importantly, I ...
2
votes
1answer
56 views

Determination of all prime numbers which give integer solution of a particular summation.

Determine all primes numbers $p$ such that $$p \sum_{k=0}^{n}\frac{1}{2k+1} \in N$$ for a given positive number $n$
3
votes
1answer
101 views

Bound of the sum $\sum_{p\le n}\frac{1}{\log(p)}$

While doing a sum I came to the sum $\displaystyle\sum_{p\le n}\dfrac{1}{\log(p)}$. Where the $\log$ is the natural logarithm. It was easy to prove that $\displaystyle\sum_{p\le ...
8
votes
2answers
327 views

How often is a sum of $k$ consecutive primes also prime?

Let's define a $k$-sum as a sum of $k$ consecutive primes. For example, $15=3+5+7$ is a $3$-sum. How many $k$-sums are themselves prime? Here's one way to formulate the question more precisely: What ...
4
votes
1answer
58 views

Linnik's theorem for kth prime in the residue class

Linnik's theorm says that for any modulus $m$, the smallest prime in a given residue class mod $m$ cannot be too large: $$ p(a,m)\ll m^L. $$ where $L$ is a constant which has been improved by many ...
5
votes
2answers
230 views

Riemann Hypothesis and the prime counting function

This article on the prime counting function mentions that the Riemann Hypothesis is equivalent to the statement $$|\pi(x)-\rm {li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657 ...
12
votes
2answers
418 views

On prime factors of $n^2+1$

It is a well-known conjecture that there are infinitely many primes of the form $n^2+1$. However, there are weaker results that one can prove. For example, There are infinitely many positive ...