All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
2
votes
0
answers
195
views
Error bounds for $\pi(x)-Li(x)$ and convergence of the associated Dirichlet integral
Following On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$, define
$$I_{s}=\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x,$$ where $\pi$ denotes the prime counting function ...
- 1,019
2
votes
0
answers
99
views
A problem in modular roots
We have three mutually coprime integers $r,t,M$ where $M\asymp K^{\frac12-2\epsilon}$ and $r,t\asymp K^{\frac14+\epsilon}$ holds with some fixed $\epsilon>0$ and $K>0$ is a large parameter. ...
- 13.9k
2
votes
0
answers
158
views
On the connection between $\pi(x)-Li(x)$ and $\theta(x)-x$
Let $\pi(x)$ be the number of primes $p$ not exceeding $x, \theta(x) = \sum_{p\leq x} \log p$ and $Li(x)$ be the logarithmic integral.
Is it true that
$$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{...
- 1,019
2
votes
0
answers
167
views
What about series involving strong primes?
I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
2
votes
0
answers
112
views
Queries on distribution of prime divisors by magnitude?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$.
What is the probability distribution of ...
- 13.9k
2
votes
0
answers
121
views
How to choose a prime p s.t. n-th cyclotomic polynomial splits into as much as possible irreducible polynomials while p is almost constant size?
The reason I ask this question is that cyclotomic polynomial is critical to the construction of lattice-based cryptography. In most of the existing lattice-based cryptographic schemes, $n$ is usually ...
- 33
2
votes
0
answers
76
views
Is there an estimate available for a sum of the form $\sum_{\mathbf{x} \equiv \mathbf{a} (H) } \mu^2(x_1 x_2)$
I am interested in a sum of the shape
$$
\sum_{ \substack{ 1 \leq x_1, x_2 \leq B\\
\mathbf{x} \equiv \mathbf{a} (H) } } \mu^2(x_1 x_2).
$$
I figured it must have been considered before, but I have ...
- 3,625
2
votes
0
answers
147
views
Skewes' number and the ratio $\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$
(A complementary post is here.)
Given the prime counting function $\pi(x)$ and the logarithmic integral $\operatorname{li}(x)$, we have Table 1,
$$\begin{array}{|c|l|}
\hline
x&\operatorname{li}...
- 12.6k
2
votes
0
answers
107
views
Maximization of product over primes
I have the following maximization problem. Let $f(p)$ be a real function on the primes, having values in $(0,1)$. Assume that $Y$ is a given (large) positive number, and that we have the bound
$$\...
- 2,207
2
votes
0
answers
149
views
$f(x)$-th largest number of prime factors
Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
- 9,114
2
votes
0
answers
392
views
How big can a set of integers be if all pairs have bounded gcd
In this recent MO question, it was shown that the maximal cardinality of a subset $A(M,N)$ of $[1,N]$ where the pairwise GCD's of all set elements are upper bounded by $M,$ with $M^2\leq N$ has size ...
- 10.4k
2
votes
0
answers
617
views
Arithmetic progression and average of two prime numbers
Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1.
$$
For all terms of $A$ greater than $\ \...
- 359
2
votes
0
answers
388
views
Relation between Maier's theorem and a conjecture of Montgomery and Soundararajan
Let us consider the number of primes in the interval $[N,N+h]$, with $h\leq N$. According to the answer given by Lucia to a previous question on the distribution of primes, it is natural to consider ...
- 965
2
votes
0
answers
292
views
Prime divisors of the difference set
Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few ...
- 23k
2
votes
0
answers
318
views
Pierpont primes
A Pierpont prime is a prime $p$ that can be written as $$p=2^u 3^v + 1.$$
What is known about Pierpont primes? I'm not a number theorist, and the best I can find is
http://en.wikipedia.org/wiki/...
- 21
2
votes
2
answers
338
views
Weak form of Brocard's conjecture
I ask this out of curiosity, motivated by a question asked by one of my students.
The Brocard's conjecture claims that there exist at least four prime numbers between $p_{i}^2$ and $p_{i+1}^2$, where ...
- 66.3k
1
vote
3
answers
845
views
How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences?
Let $ N_\chi(\alpha,T)$ be the number of zeros of $L(s=\sigma+it,\chi) = \sum \frac{\chi(n)}{n^s}$ where $c > 0$ and $(\sigma,t) $ are in the rectangle $ [\alpha,1] \times [-T,T]$.
In various ...
- 22.8k
1
vote
1
answer
317
views
An explicit value for a bound proof
I saw a proof that $|p_n - li^{-1}(n)| \leq n e^{-c \sqrt{\ln(n)}} $,
without saying anything about $c$ !
My questions is, what the explicit value of $c$ ??
It just says for some number $c$ without ...
user95470
1
vote
1
answer
172
views
Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ squares and $b$ primes with $a+b\leq 3$?
This question is related to https://math.stackexchange.com/questions/3710032/conjecture-all-but-21-non-square-integers-are-the-sum-of-a-square-and-a-prime.
We know since Lagrange that every natural ...
- 6,984
1
vote
2
answers
182
views
Prime factors bounded by $k$
Let $S$ be the set of integers with largest prime factor bounded by a given positive integer $k$. Is there a formula for the asymptotic density of such a set $S$?
- 397
1
vote
2
answers
326
views
About Omega prime function
Let $ω(n)$ be the number of distinct prime factors of $n$.
Is the inequality $ω(n)\leq C\log\log(n)$ true and if so what is the value of the constant $C$ ?
- 31
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
1
answer
314
views
Inequality of three prime factors of $x^2-1$
This is about my experimental math observation on prime factorization of $x^2-1$
We can see, for many $x\in \mathbb{Z}_+$, the expression $x^2-1=(x-1)(x+1)$ gives result as a product of twin prime. ...
- 599
1
vote
1
answer
550
views
Proof that $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log {p_n}$ [closed]
I cannot find a proof of this theorem. May anyone assist?
$p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log{p_n}$
- 121
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} ,$$
where $p(i)$ denote the $i$-th ...
user95470
1
vote
4
answers
1k
views
Distribution of composite numbers
I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :https://math.stackexchange.com/questions/1272431/distribution-of-composite-numbers
...
- 787
1
vote
1
answer
343
views
What is a non-trivial upper bound on the $k$th prime below a given prime $p$?
Given a prime number $p_0$, by Bertrand's postulate we know that
\begin{gather}
p_1\ge\frac{p_0}{2}\\
p_2\ge\frac{p_1}{2}\ge\frac{p_0}{2^2}\\
\vdots\\
p_k\ge\frac{p_0}{2^k}
\end{gather}
where $p_1,p_2,...
- 113
1
vote
1
answer
130
views
Prime inequality $P_{n.m}$ $<$ $P^{m}_n$ [closed]
Let $P_n$ be nth prime, where $n,m \in$ $\mathbb{N}$, $n,m >1$.
How i can show that $P_{n.m}$ $<$ $P^{m}_n$.
I tried to used $PNT$ but that feels like overkill, is this inequality already ...
1
vote
2
answers
233
views
Inquiry on the Chebyshev $\theta$ function
Let
$$\theta(x)=\sum_{p\leq x} \log p$$ be the Chebyshev function over primes $p$.
Computational evidence seems to suggest that $\theta(x) < x$ for every sufficiently large $x$.
But is it true ?...
- 13
1
vote
1
answer
327
views
Symmetry in Hardy-Littlewood k-tuple conjecture
Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...
- 6,984
1
vote
1
answer
385
views
Convergence of a double sum involving prime numbers
This has been moved from math.stackexchange;
I am attempting to prove/disprove convergence of the following sum
$$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p \left\{\frac{...
- 1,549
1
vote
1
answer
601
views
reference on Dirichlet theorem on primes in arithmetic progression
I appreciate if you could help me to find a reference (and a proof).
Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...
- 934
1
vote
1
answer
374
views
$\{ x/p\} $ on average
This is a vague question: Lemma 2.2 of Friedlander and Lagarias' "On the distribution in short intervals of integers having no large prime factor" says that $$\sum _{p\leq w}\left (\{ x/p\} -...
- 1,381
1
vote
1
answer
365
views
Equation of the Chebyshev $\psi$ function
Consider $\Psi(x)$ to be the Chebyshev function given by
$$\Psi(x)=\sum_{n\leq x} \Lambda(n)$$
where $\Lambda(n)$ is the Mangoldt function which is equal 0 unless $n $ is prime power, and let $(E)$ ...
- 135
1
vote
1
answer
153
views
Specializing non-trivial primality tests
Primes $p$ are integers with no factors (composite allowed) in $[1,p]$. There is a polynomial time test for them.
Given an interval $[a,b]$ what is the best way to test given integer $q$ has no ...
- 13.9k
1
vote
1
answer
729
views
Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer? [closed]
I have checked in Andrew Odlyzko, Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I ...
1
vote
1
answer
231
views
An estimation of $p_n$
There seems to exist an asymptotic line
$$\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 ...
- 862
1
vote
1
answer
199
views
Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical Bases"
I was reading Professor Nathanson's graduate texts in mathematics 164: "Additive Number Theory The Classical Bases" (http://www.alefenu.com/libri/nathansonbases.pdf), and I was wondering if ...
- 13
1
vote
1
answer
131
views
Consecutive non-powerful integers
Pair of sequences $\ v_n\ $ and $\ U_n\ $ of integers start as in the following table:
[\begin{array}{rrrrrrrrrr}
n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \ldots \\
...
- 4,786
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
$$
\pi_{2k}(x) \sim c_{2k}\int_2^x\...
- 9,114
1
vote
1
answer
271
views
Least simultaneous quadratic non-residue
Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...
- 671
1
vote
1
answer
204
views
Behavior of a quantity related to Fermat's 4n + 1 Theorem
One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to ...
- 3,699
1
vote
2
answers
955
views
Numbers of a different order?
Let $d_r$ be a divergent series of positive terms and let $s_r = \sum_{i=1}^{r}d_r$. We are interested in the sequence of numbers $S_{d_r} = s_1, s_2, \ldots$. For example if $d_r = 1/r$ the $s_r = ...
- 459
1
vote
2
answers
383
views
Is there any way to estimate this functions: $f(n)=\sum_{d|n}d\varphi(d)$ and $g(n)=\sum_{d|n}\frac{\varphi(d)}{d}$?
Let that $n$ be a natural number and $\varphi(n)$ be the Euler totient function. Is there any formula or estimation for computing functions $f,g$ such that:
$$
f(n)=\sum_{d\mid n}d\varphi(d)
$$
and
$$
...
- 139
1
vote
1
answer
286
views
GRH and the Euler product
Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
- 565
1
vote
1
answer
124
views
why is $R := \mathbb{Z}p +XF[[X]]$ an almost valuation domain?
Recall that an integral domain $R$ with quotient field $K$ is
an almost valuation domain if for every $0 \not= x \in K$, there is a positive integer
$n$ (depending on $x$) such that $x^n \in R$ or $x^{...
- 147
1
vote
1
answer
153
views
Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?
I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:
For $n, x \ge 2$ be two integers then:
$$P_{2n} \ge 2P_n$$
and
$$\pi(2x) \le 2\pi(x)...
- 1,776
1
vote
1
answer
466
views
Proof of prime gap bound? [closed]
In another question on mathoverflow (What is the best currently proven bounds on prime gaps?) the following bound on the prime gap was quoted:
$G(X)\ll \frac{X^{0.525}}{\log X}$
How do you prove this, ...
- 45
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
1
vote
1
answer
177
views
Arithmetic progressions, given a prime
I have recently become interested in reading a little more on certain directions regarding primes in arithmetic progressions (AP). I would appreciate specific paper references (with the journal and ...
user137748