All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
3
votes
0
answers
188
views
Riemann's explicit formula for square-free numbers
We know that for $x$ being a half-integer $$\sum_{n\leq x}\Lambda(n)=x-\sum_{\rho}\frac{x^\rho}{\rho}+O(1).$$
Is there a similar formula for $\sum_{n\leq x}\mu(n)^2$ in the literature? The underlying ...
- 3,062
6
votes
2
answers
571
views
Local density of numbers not divisible by small primes
Since$$\prod_{p \leq n} \left(1-\frac{1}{p}\right) =\frac{ e^{-\gamma}+o(1)}{ \log n},$$ by Mertens theorem, the density of integers in $$(X^{\theta},X],$$ which aren't divisible by primes $$p \leq X^{...
- 10.4k
3
votes
0
answers
334
views
Conditional proof of ternary Goldbach
This is a reference request.
I know that Hardy and Littlewood gave a proof of the ternary Goldbach for sufficiently large odd integers under the assumption of GRH.
Is there a modern account of ...
- 66.3k
8
votes
3
answers
1k
views
Asymptotics for primality of sum of three consecutive primes
We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where $\{p_i\}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc..
The first few values of $R_n$ are:
10, 15, 23, 31, 41, 49, 59, ...
- 105k
2
votes
1
answer
311
views
congruence for modular forms coefficients
Let $A$ be the set of modular forms $f=\sum a(n)q^n$ of weight $k$ on $\Gamma_0(N)$ with character $\chi$ whose coefficients $a(n)$ are in the ring of integers $\mathcal{O}$ of a fixed number field $F$...
- 26k
4
votes
1
answer
464
views
Odd Chebyshev, part 2
Let
$$ \forall_{n=1\ 2\ \ldots}\quad I(n)\ :=\
\frac {(6\cdot n-3)!!}{(2\cdot n-1)!!\cdot(4\cdot n-3)!!} $$
Let $\ M(n)\ $ be the smallest natural number such that
$$ M(n)\cdot I(n)\ \...
- 7,500
0
votes
0
answers
164
views
Sequences that "capture their primes"
Let $g : \mathbb{N} \rightarrow \mathbb{N}$ be an increasing function, and consider the sequence $Y = \{y_n\}$ given by $y_n = g(n)$. Let $F$ be an irreducible binary form of degree $d \geq 2$ with ...
- 26.9k
0
votes
1
answer
187
views
Expressing odd numbers as a prime minus $a^2+a$
I am looking for results about expressing odd numbers in the form
$$p-a^2-a,$$
where $p$ is a prime and $a$ is a positive integer.
Assuming Bunyakovsky conjecture this is easy as $x^2+ x+c$ are ...
- 105k
3
votes
0
answers
135
views
The probability of having $\omega(z_1\dotsb z_\kappa)<\kappa$
The problem below is suggested by this one.
Suppose that we are given $2k$ integers $x_1,\dotsc,y_k$, and we want to find an integer $a$ so that $\gcd(a+x_i,a+y_i)>1$ for each $i\in[1,k]$. This ...
CommunityBot
- 1
0
votes
1
answer
370
views
prime counting function pi bounds [closed]
is it true that for some integer $n_0$, that all integer numbers n such that $n \geq n_0$ the following holds true for the prime counting function :
$\frac{x}{\ln x} (1+\frac{1}{\ln x}+\frac{2}{\ln^2 ...
- 2,803
3
votes
1
answer
186
views
A sieve with two parameters
I am in need of a (relatively) general sieve with two parameters $y, z$. I am quite sure that on the literature there must be some result of the kind that I have in mind, probably a corollary of the ...
CommunityBot
- 1
5
votes
1
answer
1k
views
Lines in image; are they significant to prime numbers if so how?
Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where $...
- 151
2
votes
2
answers
748
views
The sequence $n^2+1$ and semiprimes
Does the set $ \{n^2+1 : n \in \mathbb N\}$ contain infinitely many semiprimes?
Also, are there any results on density of semiprimes in this sequence?
- 17.6k
2
votes
1
answer
332
views
Upper bound for product of exponents of prime factorization
Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example,
$$p(5184)=p(2^{6}\cdot 3^{4})=24,\qquad
p(65536)=p(2^{16})=16.$$
Is $p(n) = O(\log^{k}(n))$ for some constant ...
- 105k
1
vote
1
answer
175
views
Sequence of consecutive gaps in prime numbers
Let $p_i$ be $i^{th}$ prime number (for example $p_4=7$). Is there $m=m(n, k)$ such that $n \leq \min \{p_{m+i}-p_{m+i-1} : 1 \leq i \leq k\}$ for every $n$ and $k$? Even, consider the problem for $k=...
- 321
3
votes
2
answers
402
views
Estimating $\sum_{p_1\cdots p_k\leq n} \frac{1}{p_1\cdots p_k}$ for various $k$
Let $$D_2(n) =\sum_{pq\leq n} 1,$$
and
$$F_2(n) =\sum_{pq\leq n} \frac{1}{pq}$$
where $p,q$ are primes. Similarly define
$$D_k(n) =\sum_{p_1\cdots p_k\leq n} 1,$$
and
$$F_k(n) =\sum_{p_1\cdots p_k\...
- 105k
0
votes
1
answer
413
views
Upper bound for the number of even numbers sum of a prime and a semi-prime not fulfilling Goldbach's conjecture
Chen Jing Run proved that every large enough even integer is either the sum of two primes or the sum of a prime and a semi-prime (that is, the product of two primes). Golbach's conjecture states that ...
- 17.6k
8
votes
1
answer
310
views
Do arbitrary $K$-twin primes exist?
Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$.
I am interested the status in a much weaker form of ...
- 1
1
vote
1
answer
203
views
Best bound on $p, p+2k$ with $k$ fixed
Given some integer $k>0$, there are $O(x/\log^2 x)$ primes $p \le x$ such that $p+2k$ is also prime. It has been conjectured at least since Hardy-Littlewood that
$$
\pi_{2k}(x) \sim c_{2k}\int_2^x\...
- 105k
4
votes
1
answer
268
views
Large prime divisors in small intervals
For my thesis I would like to find integers (lying in a certain residue class) in small intervals which have large prime divisors. And for some reason I decided that I want all bounds appearing in my ...
CommunityBot
- 1
3
votes
2
answers
481
views
Two equivalent statements about primes
Regarding to our hypothesis in https://math.stackexchange.com/questions/1918406/a-hypothesis-about-the-conjecture-every-even-number-is-the-difference-of-two-p , we guess that the following statements ...
CommunityBot
- 1
17
votes
3
answers
3k
views
A variant of the Goldbach Conjecture
I am asking if this variant of the weak Goldbach Conjecture is already known.
Let $N$ be an odd number. Does there exist prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can ...
- 105k
0
votes
2
answers
317
views
On a coprime generalization of Cramer's conjecture
Given a large enough integer $n\in\Bbb N$ and a real $r\in\big(0,\frac12\big]$ and $n_1\in\Bbb N_{> n}$ is the smallest integer such that $n_1=AB$ for two coprime integers $A$ bigger than but close ...
CommunityBot
- 1
4
votes
1
answer
647
views
Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?
http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least one ...
CommunityBot
- 1
5
votes
1
answer
436
views
Even sharper upper bound for prime product?
In Dusart papers he proves that $\prod \limits_{p \leq x} \frac{p_i}{p_i-1} \leq e^\gamma \ln (x) \left(1+\frac{0.2}{\ln ^2 (x)} \right)$ for large numbers.
What I am asking is could we make the ...
- 17.6k
7
votes
1
answer
1k
views
Heuristic for Montgomery's conjecture
This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
...
CommunityBot
- 1
6
votes
1
answer
665
views
On the distribution of roots modulo primes of an integral polynomial
For motivation and related questions, see below.
Rough sketch of the question.
View $\bigsqcup_{p \text{ prime}} (\mathbb{Z}/p\mathbb{Z})$ as a ‘subset’ of the unit circle, via $a\pmod{p} \mapsto e^{...
- 83
0
votes
1
answer
437
views
Mertens' 3rd theorem, upper bound
Is it true that
$$\prod_{p\le x}\frac p{p-1}\le e^\gamma\ln x\left(1-\frac{0{.}011}{\ln x}+\frac{0.2}{(\ln x)^2}\right)$$
for all $x>25\,000$, where the product is over prime $p$?
- 105k
1
vote
3
answers
378
views
Is this product, involving consecutive primes, always less than or equal to $1$?
i don't know how to write math in Latex so i will try to explain it simply,
if we multiply
$$\frac{p(i)^2}{p(i)^2-1}\prod_{j=1}^5\frac{p(i+j)^2-1}{p(i+j)^2} ,$$
where $p(i)$ denote the $i$-th ...
- 13k
6
votes
3
answers
440
views
A divergent series related to the number of divisors of of p-1
Let $d(n)$ denote the number of divisors of $n$. Is it known that the series
$$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$
diverges?
This would follow immediately from the Sophie Germain Conjecture. ...
- 5,470
10
votes
0
answers
740
views
Implications of divergence of $1/\zeta(s) $ at 1/2
$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.
This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s&...
- 17.6k
4
votes
0
answers
176
views
Are there any results about this higher degree Titchmarsh divisor problem?
Does there exist an asymptotic formula for $\sum_{p\le x}\tau(p−1)^n$ ? Here $n$ is an arbitrary positive integer and $\tau$ is the divisor function. The case of $n=1$ was done by Linnik, but when $n$ ...
- 105k
2
votes
1
answer
743
views
Counting number of primes that split completely in a number field
Let $L=\mathbb{Q}(\zeta_r , a^{1/s})$ where $s|r$. Note that it is a splitting field of $f(x)= x^r-a^{r/s}$ over $\mathbb{Q}$ and thus a Galois extension of $\mathbb{Q}$.
I want to estimate : $$\pi_L(...
- 400
7
votes
1
answer
1k
views
Results on the largest prime factor of $2^n+1$
A work of Cameron Stewart (the paper has appeared in Acta Mathematica), proving a conjecture of Erdos, Stewart shows that
the largest prime factor of $2^n-1$ is at least
$n \times \exp\Big( \frac{\...
- 4,746
1
vote
1
answer
231
views
An estimation of $p_n$
There seems to exist an asymptotic line
$$\displaystyle a+bx\sim \frac{x e^x}{p_{n}-x e^x}\; ,\;n=\lfloor e^x\rfloor\tag1$$
Which suggests an estimation
$$\displaystyle g(n)=\Big(1+\frac{1}{a+b\ln ...
CommunityBot
- 1
5
votes
2
answers
314
views
Congruences for the non-divisors of Euler's $\phi(n)$
If $n$ is composite, then $\phi(n) < n-1$: hence, there is at least one number $d$ which does not divide $\phi(n)$ but divides$(n-1)$. We shall call $d$ the totient divisor of $n$. The purist will ...
CommunityBot
- 1
6
votes
6
answers
2k
views
Sequences equidistributed modulo 1
Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
I....
CommunityBot
- 1
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
$$
\sum_{r=1}^{m} \frac{...
- 43.7k
8
votes
1
answer
838
views
Density of prime pairs whose gap is less than the average gap
By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
CommunityBot
- 1
2
votes
2
answers
412
views
Asymptotic estimate of the probability of $(n, P(\sqrt{x})) \leq x$?
Let $P(x)$ be the product of all primes less or equal to $x$. The probability of $(n, P(\sqrt{x})) \leq x$ for an arbitrary $n$ is then given exactly by
$$
\prod_{p\mid P(\sqrt{x})}{\left(1-\frac{1}{p}...
- 109k
15
votes
1
answer
901
views
Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?
Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...
- 43.7k
1
vote
0
answers
165
views
Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$
Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...
- 3,625
2
votes
1
answer
441
views
Density of Sophie Germain $3\bmod 4$ primes
Is there any reason to expect the density of primes $p$ such that $2p+1$ is also a prime where $p=3\bmod4$ holds would be different from case of $p=1\bmod4$?
What if $2p+1$ is replaced by $2p-1$ and ...
- 2,283
8
votes
1
answer
243
views
Main term in the number of sign changes of $\psi(x) - x$
Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$.
Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1].
But perhaps that estimate is too crude. ...
- 43.7k
2
votes
1
answer
1k
views
What is the best currently proven bounds on prime gaps?
I did some digging around on the internet but I found tons of different equations on both lower and upper bounds for the largest possible prime gap g(n). I was wondering what are currently the best ...
- 105k
7
votes
1
answer
238
views
Density of prime divisors of $a^n + b$
Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density?
- 20.4k
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 ...
- 7,217
3
votes
3
answers
1k
views
Primes in arithmetic progressions in number fields
My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...
CommunityBot
- 1
2
votes
0
answers
149
views
$f(x)$-th largest number of prime factors
Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
- 9,114
19
votes
1
answer
2k
views
How many primes can there be in a short interval?
Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
\limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \...
- 114k