All Questions
48 questions
29
votes
7
answers
7k
views
Asymptotic density of k-almost primes
Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is
$$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\...
- 9,114
20
votes
1
answer
594
views
Distinct exponents in the factorization of the factorial, a problem of Erdős
In the 1982 paper below, Paul Erdős proved that if $h(n)$ is the number of distinct exponents in the prime factorization of $n!$ then $$c_1\Big(\frac{n}{\log n}\Big)^{1/2} < h(n) < c_2\Big(\frac{...
- 203
11
votes
2
answers
1k
views
Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
11
votes
1
answer
324
views
Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
- 5,470
8
votes
4
answers
2k
views
Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)
What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)?
In other words, I am curious about the state of the ...
- 1,329
6
votes
1
answer
349
views
Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$
To begin, let us set
$$A_Q(n):=\sum_{d|n \\ d<Q}\mu(d)$$
If we fix $Q$ and let $n$ vary, we get a very surprising amount of cancellation. For instance, the trivial bound
\begin{align*}
\mathbb{E}_{...
- 2,902
5
votes
2
answers
641
views
Asymptotic Distribution of Primes
Given an integer $n$ and let $1\leq m\leq n$ be such that $n$ and $m$ are coprimes define
$$
\mathcal{N_{n,m}}:=\text{the set of primes $p$ such that $p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)$}.
$$
...
- 3,626
5
votes
4
answers
820
views
Can one show combinatorially how $\operatorname{lcm}(1, \dotsc, n)$ grows?
Let us write $M(n)$ for $\operatorname{lcm}(1,\dotsc,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, ...
- 13k
5
votes
0
answers
355
views
What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?
What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...
- 5,470
4
votes
2
answers
704
views
Estimate related to the Möbius function
I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less or equal than $x$ that are square free and with exactly $k$ primes on its decomposition. ...
4
votes
1
answer
628
views
Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$
Let us define the following functions:
\begin{equation*}
\small A(x)=\prod_{\substack{p\leq x\\ p\equiv 3 \bmod 4}} \Big(1-\frac{1}{p}\Big), \mbox{ } \mbox{ }
B(x)=\prod_{\substack{p\leq x\\ p\...
- 3,098
4
votes
1
answer
549
views
Sum over reciprocal of primes times coefficient
I would like to show that
$$
\sum_{p\leq x} \frac{1}{p^{1+2/\log x}}\left(\frac{\log\left(x/p\right)}{\log(x)}\right)^2=\log\log x +\mathcal{O}(1)
$$
What I have tried
Since we know that
$$
\sum_{p\...
- 199
4
votes
2
answers
472
views
Sharp estimates for Meissel-Mertens constant
I wondered if it is possible to get a similar inequality like $(1.1)$ of Michael D. Hirschhorn, Approximating Euler's Constant, The Fibonacci Quarterly, Volume 49, Number 3 (August 2011) for the ...
4
votes
1
answer
235
views
Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number
Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
- 5,470
4
votes
1
answer
954
views
Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture
The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambda(m)=\...
- 1,183
4
votes
2
answers
891
views
Asymptotics for the number of ways to sum primes such that the sum is <= n
Hello!
Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq \...
- 3,463
4
votes
1
answer
291
views
A similar lemma to a lemma due to Lagarias, for the partial sums of reciprocal of primes
I was inspired in Lemma 3.1 of [1] and in the Theorem 4.12 of [2] to ask about a similar statement that shows Lagarias in his paper as Lemma 3.1.
The Lemma from Lagarias's paper is that if $H(n)=\...
4
votes
1
answer
434
views
Quadratic progressions with very high prime density
In my previous MO question (see here), I solved the case for arithmetic progressions $f_k(x)=q_k x+1$. The solution is this:
The list of sequences $f_k(x)$, each one corresponding to a specific
$k$, ...
- 3,098
4
votes
0
answers
263
views
Asymptotic number of "modular primes"
We can say that a number $p$ is prime modulo $N$ if for any two numbers $1<a,b<p$, $ab \not\equiv p \pmod N$. We will define $p(n)$ to be the number of primes mod $n$. I'm wondering about the ...
- 3,319
4
votes
0
answers
412
views
Effective prime number theorem
The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...
- 13.9k
4
votes
0
answers
306
views
Effective version of the Bombieri-Vinogradov theorem
Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?
- 1,890
3
votes
2
answers
818
views
Density of the set of numbers whose sum of digits is prime
Let $A$ be the set of numbers whose sum of digits is prime (http://oeis.org/A028834).
I would like to know if $A$ has zero natural density, that is, if $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...
- 41
3
votes
2
answers
218
views
The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\operatorname{ali}(n)$ the inverse of the logarithmic integral
I'm inspired in [1] to ask the following question. My problem is that I have not an implementation of the inverse of the logarithmic integral $\operatorname{Li}(x)=\int_2^x\frac{dt}{\log t}$, that ...
3
votes
1
answer
205
views
How does the minimal degree of a monic polynomial with all values divisible by $p^n$ asymptotically behave?
Let $p$ be a prime number. For every $n \in \mathbb N$, let
$A_{p,n}:=\{\deg P(X) : P(X)\in \mathbb Z[X]$ is monic and $p^n|P(m), \forall m \in \mathbb Z$ $\}$ .
As user abx notes below, $A_{p,n}$ ...
user111492
3
votes
2
answers
876
views
Asymptotics for the number of partitions of $n$ into odd prime parts
Hello!
I am interested in the asymptotic behavior of the function $p_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - http://oeis.org/A099773 .
I couldn't find any ...
- 3,463
3
votes
1
answer
348
views
On conjectures about the arithmetic function that counts the number of Sophie Germain primes
I've edited this post two years ago on Mathematics Stack Exchange, with identifier 3590406 and same title On conjectures about the arithmetic function that counts the number of Sophie Germain primes, ...
3
votes
1
answer
516
views
About the asymptotics of LCM
Let $g(x,c)$ be a uniformly random integer in the range $(x,x+c)$ and $LCM[x_1,x_2...x_i]$ the lowest common multiple of the integers $x_i$.
A) Does the limit of (the asymptotics of $LCM[g(3^1,c),g(...
- 61
3
votes
0
answers
151
views
On an inequality for the arithmetic function counting the number of primes $\lfloor n^c\rfloor$ in the spirit of Ramanujan's prime counting inequality
In page 3 of [1] (please see if you need it the book by Berndt) Axler refers an inequality that involves the prime-counting function $\pi(x)$ and that was deduced by Ramanujan. I'm curious to know if ...
3
votes
0
answers
320
views
On sets of coprime numbers
We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by $...
user76479
2
votes
1
answer
740
views
Does the Riemann hypothesis predict a bound for this prime-counting function?
Does the Riemann hypothesis predict an upper bound for
$$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$
where
$$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
- 1,126
2
votes
3
answers
366
views
Is this number theoretic quantity bounded above?
I am considering a combinatorial argument which involves the following quantity. We use the prime counting function $\pi(n)$ and to save on exponents we set $h=\pi(n/2)$. The quantity as a function ...
- 13k
2
votes
2
answers
555
views
What is the growth rate for divisibility of integers
For a natural number $n\geq 1$, let $PF(n)$ denote the number of prime factors (with multiplicity) of $n$. For example, since $48=2*2*2*2*3$, we have $PF(48)=5$.
For any natural number $N\geq 1$, ...
- 8,659
2
votes
0
answers
136
views
Average length of consecutive integers which have an increasing number of divisors
Consider the nine consecutive natural numbers starting from $1584614377$.
...
- 5,470
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
2
votes
0
answers
203
views
Asymptotics on a double sum over primes
I am attemping to find asymptotics of
$$\sum_{p \leq n}\ln p \left( \sum_{k=1}^\infty \left(\left\{\frac{n}{p^k} \right\} - \left\{\frac{n-1}{(p-1)p^k} \right\} \right) - \left\{\frac{n-1}{p-1} \...
- 1,549
2
votes
0
answers
167
views
What about series involving strong primes?
I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
1
vote
1
answer
867
views
$n$th prime: a better approximation
Let $p_n$ be the $n$-th prime, then from Wikipedia I got that
$p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.
What is a ...
user95470
1
vote
2
answers
173
views
A distribution related to Fermat's two squares theorem
Fermat's two squares theorem tell us that every prime number $p \equiv 1 \pmod 4$ can be written in a unique way as $p = a^2 + b^2$ for two positive integers $a < b$. In particular, we can ...
- 33
1
vote
1
answer
141
views
Compare $\operatorname{rad}(an+b)$ and $\varphi(cn+d)$ in a simple and interesting inequality, for some choice of integers $a,b,c$ and $d$
We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...
user75829
1
vote
1
answer
230
views
Asymptotic for a number theoretic sequence and its Dirichlet series' convergence
I would like to know the asymptotic behaviour at large $n$ for $t\in\mathbb{R}$, $t\neq0$ of the following function:
\begin{align*}
A_n(t)&=\sum_{q=\frac{a}{b}\in \mathbb{Q}^+|\gcd(a,b)=1 \& ...
- 634
1
vote
0
answers
315
views
From an inequality for the Euler's totient function to a combination of Firoozbakht's conjecture and Nicolas' criterion for the Riemann hypothesis
In this post we ask about the veracity of an inequality deduced from a combination of Firoozbakht's conjecture (see [1] or [2]) and Nicolas' criterion for the Riemann hypothesis (see for instance [3])....
1
vote
0
answers
200
views
Limit of Sequence of unusual Prime Product
Let $p_n$ be the nth prime and $p_L$ be closest to its square root:
\begin{equation}
p_L^2 \approx p_n \approx x
\end{equation}
Let $\sigma \in Z^+$ be a positive integer constant. Define the ...
0
votes
1
answer
404
views
Asymptotics of "ugly" function elucidate Goldbach's conjecture?
Question
We now define the following "ugly" function:
$$ A_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly"...
- 89
0
votes
1
answer
204
views
On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s})$ as $X\to\infty$
Few weeks ago an user from Mathematics Stack Exchange answered my question On an inequality that involves products and sums related to the sequence of semiprimes (asked May 26). It seems that for ...
0
votes
1
answer
669
views
Paradox in additive combinatorics
Let $S$ be an infinite set of positive integers. Let us define the following quantities:
$N_S(z)$ is the number of elements of $S$, less or equal to $z$
$r_S(z)$ if the number of positive integer ...
- 3,098
0
votes
0
answers
68
views
Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
0
votes
0
answers
80
views
Relevance of the deduction of similar theorems than Maier's theorem for other prime constellations
A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
0
votes
0
answers
89
views
A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...