0
votes
2answers
48 views

Convergence of $\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$

Does this diverge or converge ?? $$\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$$ where $H_n$ is the nth harmonic number, $p_n$ is the nth prime. My impression is that it diverges, but I don't ...
0
votes
1answer
72 views

How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?

Recently I am reading the textbook of Apostol, Mathematical Analysis, Second Edition. On page 7, there is a theorem 1.10: If $n$ is a positive integer with is not a perfect square, then $\sqrt{n}$ is ...
1
vote
3answers
90 views

$x^2-p=0$, with $p$ prime, have irrational roots.

Unaware that $\sqrt{p}$ is irrational, prove that as $x^2-p=0$ have irrational root for $p$ prime? How would you use the criterion of Eisenstein?
1
vote
3answers
74 views

Proof that the derivative of the prime counting function is the probability of prime?

The derivative of the estimation of the prime counting function, $\frac{x}{ln(x)}$, is $\frac{ln(x)-1}{ln(x)^2}$, which is approximately $\frac{1}{lnx}$ for large values of $x$. According to the prime ...
2
votes
0answers
31 views

$\sum_{i=1}^{k}[\sqrt{ip}]=\frac{p^2-1}{12}$, $p$ is a prime of the form $4k+1$ [duplicate]

$\sum_{i=1}^{k}[\sqrt{ip}]=\frac{p^2-1}{12}$, $p$ is a prime of the form $4k+1$ How to prove this?
7
votes
1answer
76 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
97 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.
1
vote
2answers
91 views

Aside from approximation functions, are there any proven functions that produce an exact $n$th prime?

Aside from approximation functions, are there any functions that produce an exact $n$th prime?
0
votes
0answers
59 views

sum over prime index done by a weird sieve?

As you might have noticed i considered in 2 previous questions sums of the form $f(p_i x)$ where the sum is over the primes $p_i$ ( between some integer bounds : $a \leqslant p_i \leqslant b$ ) , $x$ ...
7
votes
2answers
208 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 ...
3
votes
2answers
150 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 ...
1
vote
0answers
127 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 ...
5
votes
1answer
217 views

Asymptotic behavior of $\sum_{i>0} x^{p_i}$ as $x \to 1^-$

The sum of natural numbers $ \sum_{n>0} x^n = \frac{x}{1-x}$, so as $x\to1^-$ it diverges as $(1-x)^{-1}$. So I wondered what would happen if we make the summation set thinner, i.e. $\sum_{n \in A} ...