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122 views

Convergence of a series related to counting distinct prime factors

I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
Zachary Hoelscher's user avatar
2 votes
0 answers
120 views

On the integer of the form p^a q^b closest to a given integer N

If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
Azoth's user avatar
  • 69
2 votes
2 answers
424 views

"Squeezing" the primes?

The logical idea here is to map a curve that encodes the primes into the region $(0,1)^2$ and analyze the distribution there more easily and achieve tight bounds. To assess the distribution of primes, ...
John McManus's user avatar
1 vote
0 answers
179 views

Getting rid of complex zeros of function with zeros the primes?

From our Note: simple real function with zeros greater than one the primes simple real function with zeros greater than one the primes: $j_1(x)=(\sin(\pi x))^2+(\sin(\pi \frac{\Gamma(x)+1}{x}))^2$. ...
joro's user avatar
  • 25.4k
4 votes
1 answer
395 views

Mertens formulas aren't enough for prime number theorem

For the primes it's true that $$ \sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x) $$ where, $M$ is suitable constant, and, moreover, the prime number theorem gives that $$ \lim_{x\to\infty}\frac{\...
user627482's user avatar
5 votes
1 answer
351 views

Is every integer $\ge 312$ the sum of two integers with triangular divisors?

We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3$, $1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a ...
Nilotpal Kanti Sinha's user avatar
11 votes
1 answer
436 views

How many numbers $\le x$ can be factorized into three numbers which form the sides of a triangle?

Note: Posting in MO since it was unanswered in MSE Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \...
Nilotpal Kanti Sinha's user avatar
5 votes
0 answers
137 views

Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?

This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO. For every ...
Nilotpal Kanti Sinha's user avatar
28 votes
3 answers
3k views

Expressing the Riemann Zeta function in terms of GCD and LCM

Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased, $$ \frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
Nilotpal Kanti Sinha's user avatar
10 votes
0 answers
268 views

On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$

I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well. For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
Lucio Tanzini's user avatar
5 votes
0 answers
89 views

Is the ratio of a number to the variance of its divisors injective?

The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
Nilotpal Kanti Sinha's user avatar
23 votes
1 answer
3k views

Does the average primeness of natural numbers tend to zero?

This question was posted in MSE. It got many upvotes but no answer hence posting it in MO. A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...
Nilotpal Kanti Sinha's user avatar
1 vote
0 answers
188 views

Questions on Riemann's explicit formula

If we consider this version of the prime-counting function $$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$ (with $\pi$ being the normal prime-counting function), then we can write $\...
tobias's user avatar
  • 749
3 votes
2 answers
491 views

Unknown bias in a distribution related to prime numbers

If $n$ is composite then $\phi(n) < n-1$, hence there is at least one divisor $d$ of $n-1$ which does not divide $\phi(n)$. We call $d$ as the totient divisor of $n$. Trvially, if $n$ is prime then ...
Nilotpal Kanti Sinha's user avatar
3 votes
1 answer
290 views

Fluctuating constants

Let $p_k$ be the $k$-th prime number, $\gamma$ be the Euler-Mascheroni constant and $M$ be the Meissel–Mertens and let $m$ be the integer part of $\log p_n$. We can show that $$ \sum_{r=1}^{m} \frac{...
Nilotpal Kanti Sinha's user avatar
2 votes
0 answers
229 views

Analytic varieties for the primes and the twin primes

I am wondering what real and complex analysis say about the primes and twin primes. According to Wikipedia analytic variety is defined locally as the set of common zeros of finitely many analytic ...
joro's user avatar
  • 25.4k
8 votes
1 answer
838 views

Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
Nilotpal Kanti Sinha's user avatar
13 votes
2 answers
2k views

Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that. $$ p_n = n \...
Nilotpal Kanti Sinha's user avatar
46 votes
4 answers
8k views

Why could Mertens not prove the prime number theorem?

We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$ where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln ...
Nilotpal Kanti Sinha's user avatar