4
votes
2answers
57 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
21 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
22 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
25 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
33 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
41 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
55 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
42 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
47 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
117 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
99 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
69 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
124 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 ...
1
vote
0answers
55 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
76 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
80 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
53 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 $n$
2
votes
1answer
76 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
287 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
50 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
161 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 ...
11
votes
2answers
297 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 ...
2
votes
1answer
62 views

On asymptotics for the number of ways to write $2n$ as the sum of two primes

Let $g(n)$ be the number of ways to write $2n$ as the sum of two primes. Define $G(n) = g(2) + \cdot \cdot \cdot + g(n)$. Are there any conjectured or known asymptotics or bounds for $G(n)$? ...
4
votes
1answer
68 views

Prove that $2^x < \prod_{p\le x} p < (13/4)^x$ for sufficiently large x

Prove that $2^x < \prod_{p\le x} p < (13/4)^x$ for sufficiently large x. Here $p$ is prime. So if we take logs we need to show for sufficiently large x, $x\log 2 < \sum_{p\le x}\log p < ...
1
vote
1answer
106 views

Ramanujan's prime counting inequality

This article says that Ramanujan claimed that $$ \pi^2(x) < {ex \over \log x} \pi(x/e), $$ and that it is indeed true for $x$ sufficiently large. They really glaze over the proof though, and I'm ...
3
votes
2answers
116 views

Proving all sufficiently large integers can be written in the form $a^2+pq$.

This is one of those numerous questions I ask myself, and to which I seem unable to answer: Can every integer greater then $657$ be written in the form $a^2+pq$, with $a\in\mathbb Z$ and $p,q$ ...
2
votes
0answers
40 views

Can anyone sketch an outline of Iwaniec's proof for the upper bound regarding the Jacobsthal function?

A proof by H. Iwaniec in 'On the problem of Jacobsthal, Demonstratio Math. 11, 225–231, (1978)' shows that: $$j(N) \ll \log \log (N)$$ where $j(N)$ is the Jacobsthal function. I am very interested ...
-1
votes
1answer
83 views
1
vote
1answer
54 views

Lower bound for second Chebyshev function

I was wondering is there any simple way to find nice lower bound for second Chebyshev function given by formula: $$\psi(x)=\sum_{p\le x} \left\lfloor \frac{\ln x}{\ln p} \right\rfloor\ln p$$ that is ...
2
votes
1answer
95 views

Probability property that the longest side of primitive Pythagorean triples is prime

If we consider the set of the first $n$ primitive Pythagorean triples, then the probability that the triple's longest side is prime is approximately $\dfrac{1}{\log_{11.475}n}$ based on Mathematica’s ...
4
votes
1answer
143 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
2
votes
1answer
72 views

Between Mertens' theorems

It is well-known that $$ \sum_{p\le x}\frac{\log p}{p}=\log x+O(1) $$ and $$ \sum_{p\le x}\frac1p=\log\log x+M+o(1). $$ What is the order of $$ \sum_{p\le x}\frac{\sqrt{\log p}}{p} $$ ?
6
votes
1answer
187 views

Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
1
vote
1answer
91 views

Prove that the number of primes satisfying this is $\log(n)$

Let $\textrm{ord}(n)$ denote the number of primes $p$ such that order of $10$ modulo $p$ is $n$. Prove that $\textrm{ord}(n) \sim \log(n)$.
0
votes
2answers
48 views

At what rate are composites removed in a set after each prime multiple is cancelled out?

I was looking at sieves today, mainly sieving for primes and I noticed a pattern type thing. As I crossed out primes in a small set, the number of composites that were crossed out decreased. I haven't ...
1
vote
1answer
48 views

Closed forms for $\lim_{x\rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$

Im looking for closed forms for $\lim_{x \rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ where $x$ is a positive real, $a$ is a given real, $p$ is the set of primes such that the ...
0
votes
1answer
65 views

To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from ...
0
votes
0answers
52 views

order of magnitude of first moment of prime's distribution?

I have these this question in mind for a while and struggling with. Question: Is it possible to define a first moment (mean) of the distribution of the primes' infinite sequence? And if this can be ...
2
votes
2answers
70 views

Growth rate of product of smallest prime factors

For $n\in \mathbb{N}$, let $p(n)$ denote the smallest prime dividing $n$. Then consider the function $f:\mathbb{N}\rightarrow \mathbb{N}$ defined by $f(n)= \prod_{k=1}^{n}p(k)$. Question: What is ...
1
vote
1answer
33 views

Proving that $\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3})$

How do I prove that $$\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3}).$$ I tried to use Prime Number theorem for $\pi(x)$ and then approximating the summation by integral, but when I used ...
10
votes
2answers
354 views

A problem about the largest prime factor of $n^2+1$

Let $f(n)$ be the largest prime factor of $n$. The image of function $g(n)=\sqrt{f(n^2+1)}$ is like this: Question: If we want to draw a horizontal line which bisects the points from $n=1$ to ...
5
votes
0answers
206 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...
1
vote
1answer
117 views

Challenging the Chebychev function / prime number theorem?

The prime number theorem accords with the following equation for the first Chebychev function that: $$\lim_{x\rightarrow\infty}\frac{\vartheta(x)}{x}=1 \qquad (1)$$ According to Muñoz García, E. and ...
5
votes
1answer
92 views

Number of prime factors of difference of two numbers

As is the custom, define $\omega(m)$ to be number of distinct primes dividing $m$. Also, let $P(m)$ represent set of primes divisors of $m$. Let $S=\{p_1,p_2,\ldots,p_n\}$ be a set of $n$ distinct ...
2
votes
1answer
100 views

The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$?

The von Mangoldt function: $$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$ establishes a ...
9
votes
3answers
213 views

Where is the fallacy in the argument using Prime Number Theorem

I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem: Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say ...
6
votes
1answer
76 views

Prove that the non-trivial root of $\sum_{k=1}^{2n} p_kx^k=0$ tends to $-1$

I looked at $$ \sum_{k=1}^{2n} p_kx^k=0, $$ where $p_k$ is the $k$th prime. I found that, next to the trivial root $x_0=0$, there is only one more root $x_n$ that tends towards $-1$, when $n$ ...
2
votes
0answers
251 views

Riemann $\zeta(s)$ non-trivial zeros on $Re(s)=1/2$ and the “spin” of the primes? [closed]

Can this conjecture be true? Let me intro before getting to the conjcture, but for those who like to go straight, see the bold paragraph at the end. Recently I was reading through the well known ...
1
vote
1answer
94 views

Proving that Bombieri's Theorem implies Linnik's theorem

I'm stuck on a line in the proof of Bombieri implies Linnik, where Bombieri: For primitive $\chi$ mod $q$ with $q \leq T$ we define $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; ...
4
votes
2answers
129 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) \;\;\; = \;\;\; ...