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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 $...
Nilotpal Kanti Sinha's user avatar
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)}{\...
Steven Clark's user avatar
  • 1,126
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 ...
Daniel Weber's user avatar
  • 3,319
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 -...
Charles Bouillaguet's user avatar
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 ...
user142929's user avatar
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 ...
user142929's user avatar
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, ...
user142929's user avatar
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 ...
user142929's user avatar
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 ...
Nilotpal Kanti Sinha's user avatar
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 $1584614377$. ...
Nilotpal Kanti Sinha's user avatar
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{...
sebaztian's user avatar
  • 203
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 ...
user142929's user avatar
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}_{...
Milo Moses's user avatar
  • 2,902
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 $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...
dache1771's user avatar
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 $k$, ...
Vincent Granville's user avatar
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\...
Vincent Granville's user avatar
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 ...
Vincent Granville's user avatar
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 ...
Vincent Granville's user avatar
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 ...
user142929's user avatar
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} \...
Brian's user avatar
  • 1,549
2 votes
3 answers
365 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 ...
Gerhard Paseman's user avatar
5 votes
4 answers
819 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, ...
Gerhard Paseman's user avatar
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])....
user142929's user avatar
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 ...
user142929's user avatar
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)=\...
user142929's user avatar
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 ...
user142929's user avatar
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' ...
user142929's user avatar
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\...
asd's user avatar
  • 199
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. ...
Martin Mansilla's user avatar
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 ...
Nilotpal Kanti Sinha's user avatar
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)=\...
Mats Granvik's user avatar
  • 1,183
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 ...
user avatar
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 \& ...
Alexandre's user avatar
  • 634
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, $A_{p,n}$ ...
user avatar
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 ...
MacNamara's user avatar
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 ...
user avatar
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 ...
Turbo's user avatar
  • 13.9k
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"...
drewdles's user avatar
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 $...
user avatar
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?
Mayank Pandey's user avatar
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 ...
pre-kidney's user avatar
  • 1,329
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 ...
Kim E Lumbard's user avatar
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 $N\geq 1$, ...
David Spivak's user avatar
  • 8,659
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(...
hakuna's user avatar
  • 61
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 \...
Jernej's user avatar
  • 3,463
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 ...
Jernej's user avatar
  • 3,463
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 $$ \mathcal{N_{n,m}}:=\text{the set of primes $p$ such that $p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)$}. $$ ...
ght's user avatar
  • 3,626
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)!\...
Charles's user avatar
  • 9,114