All Questions
16 questions
2
votes
0
answers
131
views
Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n$...
- 3,098
1
vote
0
answers
63
views
Set from a diophantine equation with similar statistics to primes
While doing some computational calculations with some diophantine equations, I came across with some sequences from solutions of quartic and quintic equations with slowly decreasing frequency, similar ...
3
votes
1
answer
474
views
Curious infinite product, convergence, connection to prime numbers
I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with ...
- 146
24
votes
1
answer
2k
views
Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
- 50.6k
4
votes
1
answer
219
views
Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$
Note: Posting in MO since it was unanswered in MSE
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
- 645
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 105k
3
votes
1
answer
134
views
Is it possible to find an estimate of $\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$?
Is it possible to find an estimate of the summation
$$s(n)=\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$$
where $\varphi(n)$ is the totient function and $p_k$ the k-th prime?
The corresponding series seems ...
2
votes
0
answers
422
views
Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
6
votes
0
answers
257
views
Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$
For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$
with $T_1=1$, where $p_k$ denotes the $k$-th prime.
So multiplying by $(-1)^n$ and telescoping gives that for ...
- 63.9k
3
votes
0
answers
183
views
From Firoozbakht's conjecture to set interesting conjectures for sequences or series of primes
In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if ...
2
votes
0
answers
167
views
What about series involving strong primes?
I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
2
votes
0
answers
156
views
Questions about a certain sequence of naturals generated by primorials
I'm working on the following sequence of naturals (which is NOT listed in OEIS)
$$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$
whose elements are generated this way
$$3=(...
- 63.9k
56
votes
1
answer
4k
views
A mysterious connection between primes and $\pi$
The Prime Number Theorem relates primes to the important constant $e$.
Here I report my following surprising discovery which relates primes to $\pi$.
Conjecture (December 15, 2019). Let $s(n)$ be ...
- 359
5
votes
1
answer
943
views
Are all the numbers $\pi(n^2)/n^2\ (n=1,2,3,\ldots)$ pairwise distinct?
For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. A well-known conjecture of Legendre states that $\pi(n^2)<\pi((n+1)^2)$ for any positive integer $n$. Here I ask the ...
- 4,713
3
votes
0
answers
153
views
The behavior of series involving special subsets of the prime numbers
It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ \...
- 19.6k
8
votes
2
answers
756
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{ GPF}(...
- 83