0
votes
1answer
34 views

Convergence of a modified sum of prime reciprocals for all $s \in \mathbb{C}$?

It is known that $\displaystyle \sum^\infty_{p \in \mathbb{P}} \frac{1}{p^s}$, with $\mathbb{P}$ the set of primes, only converges for $\Re(s) > 1$. The following sum of primes seems to converge ...
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 ...
1
vote
4answers
527 views

The n-th root of a prime number is irrational

If $p$ is a prime number, how can I prove by contradiction that this equation $x^{n}=p$ doesn't admit solutions in $\mathbb {Q}$ where $n\ge2$
0
votes
2answers
101 views

Nonexistence of Limit of Sum of Prime Factors

In trying to prove the following problem, I find great difficulty in proceeding to generalizing some results: Let $s(n)$ be the sum of prime factors of an integer $n$. Prove that $\lim_{n \to \infty} ...
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 ...
4
votes
2answers
106 views

Real numbers as infinte product of primes

We can uniquely write every number in $\mathbb{Q}_+$ as $\prod_{i=1}^{N} p_i^{n_i}$ where $p_i$ is the $i$th prime number and $\{ n_i \}_{i=1}^{N}$ is some finite sequence of indices, with each $n_i$ ...
3
votes
1answer
71 views

Bertrand's postulate proof

Regarding http://michaelnielsen.org/polymath1/index.php?title=Bertrand%27s_postulate I think the last inequality should be $4^{n/3}\le(2n+1)(2n)^{\sqrt{2n}}$. But even when the RHS is decreased from ...
5
votes
2answers
205 views

Primes and the Unit circle.

Consider the "prime spiral" $f(z) = \sqrt{z}\exp(2\pi i \sqrt{z})$, for integer $z$. It has been shown that the intersections of $f$ with some quadratic curves contain a significantly disproportionate ...
2
votes
1answer
65 views

A passage in the newman proof of the prime number theorem.

In the proof of the statement that $\theta(x) \sim x$ based on the fact that $\int_1^\infty { \frac{\theta(x) - x}{x^2}dx } < \infty$ We assume that for some $\lambda > 1$ there are ...
8
votes
1answer
271 views

A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$

Let $ p_1 <p_2 <\ldots <p_k <\ldots $ the increasing list in set $\mathbb{P}$ of all prime numbers . It is well known (by Infinite geometric series $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$ ...
26
votes
2answers
658 views

A beautiful limit involving primes and composites

I observed the following limit empirically. Let $p_n$ be the $n$-th prime and $c_n$ be the $n$-th composite number then, $$ \lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}\frac{p_n c_n}{p_n c_n + ...
7
votes
1answer
144 views

Euler's proof for the infinitude of the primes

I am trying to recast the proof of Euler for the infinitude of the primes in modern mathematical language, but am not sure how it is to be done. The statement is that: $$\prod_{p\in P} ...
7
votes
1answer
72 views

Proving $\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$?

$$\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$$ $P$ is primes. Interesting question ran across while tutoring. ...
6
votes
1answer
94 views

Testing for convergence $\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$

How would we test for convergence the series below? $$\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$$ where $p_i$ is the $i$th prime number. I'd be glad to learn an elementary way. Thanks.
3
votes
2answers
143 views

A series with prime numbers and fractional parts

Considering $p_{n}$ the nth prime number, then compute the limit: $$\lim_{n\to\infty} \left\{ \dfrac{1}{p_{1}} + \frac{1}{p_{2}}+\cdots+\frac{1}{p_{n}} \right\} - \{\log{\log n } \}$$ where $\{ x ...
0
votes
1answer
86 views

Is there a forumla for number of primes preceding a natural number?

I am guessing there is no known analytical function which gives such a formula. This question came to mind while attempting a rather basic proof. I am trying to show that the number of primitive ...
5
votes
1answer
209 views

Sum of alternating reciprocals of logarithm of 2,3,4…

How to determine convergence/divergence of this sum? $$\sum_{n=2}^\infty \frac{(-1)^n}{\ln(n)}$$ Why cant we conclude that the sum $\sum_{k=2}^\infty (-1)^k\frac{k}{p_k}$, with $p_k$ the $k$-th ...
25
votes
1answer
506 views

Are sines of primes dense in $[-1,1]?$

Let $P$ be the set of all prime numbers. Is $\sin(P)$ dense is $[-1,1]?$ How could we approach such a problem?
1
vote
1answer
190 views

Prime reciprocals sum

Let $a_i$ be a sequence of 1's and 2's and $p_i$ the prime numbers. And let $r=\displaystyle\sum_{i=1}^\infty p_i^{-a_i}$ Can r be rational, and can r be any rational > 1/2 or any real ? ver.2: ...
4
votes
0answers
424 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
3
votes
2answers
298 views

Minimal dense subset of $\mathbb{Q} \cap [0,1]$

The following question was a problem in an Analysis exam: Let $n \in \mathbb{N}$. Define $A_{n} := \displaystyle \left\{\frac{k}{2^n} \bigg| k \in \mathbb{Z}, 0 \leq k \leq 2^n \right\}$. Let ...
6
votes
2answers
594 views

Does the sum of reciprocals of primes converge?

Is this series known to converge, and if so, what does it converge to (if known)? Where $p_n$ is prime number n, and $p_1 = 2$, $$\sum\limits_{n=1}^{\infty} \frac{1}{p_n}$$
8
votes
1answer
425 views

Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$

Consider the closed interval $[0,1]$, there is $\frac{2}{3} \in [0,1]$ where $p=2$ and $q=3$. Similarly consider $[2,3]$, one can have $\frac{5}{2} \in [2,3]$ where $p=5$ and $q=2$. Does every ...