# Tagged Questions

**2**

votes

**3**answers

291 views

### Does this 'alternating' Euler product converge for all $\Re(s) > 0$?

Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...

**10**

votes

**1**answer

495 views

### Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum?
$$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{n}} ...

**13**

votes

**2**answers

704 views

### An Euler-proof that cannot be repaired?

Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely ...

**2**

votes

**1**answer

418 views

### To express $e^{\sum \limits_{k=0}^\infty q^{2^k}}$ as product terms of $(1-q^k)^{c(k)}$

$|q|\lt1$
$A(q)=\sum \limits_{k=0}^\infty q^{2^k}$
Easily We can see that
$$A(q)=q+A(q^2)\tag 1$$
Let's assume we redefine $A(q)$ as below
$A(q)=-\sum \limits_{k=1}^\infty c_k \ln{(1-q^k)}$
I ...

**1**

vote

**0**answers

128 views

### Limit of Sequence of unusual Prime Product

Let $p_n$ be the nth prime and $p_L$ be closest to its square root:
\begin{equation}
p_L^2 \approx p_n \approx x
\end{equation}
Let $\sigma \in Z^+$ be a positive integer constant. Define the ...

**7**

votes

**2**answers

489 views

### The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$

Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...

**5**

votes

**0**answers

686 views

### Convergent series of primes [closed]

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 ...

**4**

votes

**2**answers

587 views

### Asymptotics for the number of ways to sum primes such that the sum is <= n

Hello!
Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq ...