All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
6
votes
1
answer
382
views
Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)
Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number
of prime numbers $p \leq n$ in the residue class $r$ (mod $m$).
Further let $1 = r_1 < r_2 < \dots < r_{\varphi(m)}...
- 19.6k
6
votes
3
answers
2k
views
Better error bounds for partial sums of reciprocals of primes?
One of Mertens' theorems gives that
$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$
where $B$ is a constant near $0.26$ in value and $E(k)$ is an error
term whose size is ...
- 13k
6
votes
1
answer
826
views
Analogue of van der Corput sequence for prime numbers
A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by placing a decimal point and ...
- 459
6
votes
1
answer
903
views
How to explain this prime gap bias around last digits?
My question is related to this article by Oliver and Soundararajan (article about a bias in the distribution of the last digits of consecutive prime numbers).
After trying some python experimental ...
6
votes
2
answers
411
views
A simultaneous generalization of the Grunwald-Wang and Dirichlet Theorems on primes
By Grunwald-Wang Theorem, if for some odd number $n$ the equation $x^n=a$ has no solutions in $\mathbb Z$, then the equation $x^n=a\mod p$ has no solutions for some prime number $p$. I am interested ...
- 41.8k
6
votes
2
answers
1k
views
The shortest interval for which the prime number theorem holds [closed]
It is well known that the prime number theorem on the form
\begin{align*}
\pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)}
\end{align*}
breaks down for short enough intervals, e.g. taking $y=(\log x)^\...
- 965
6
votes
1
answer
727
views
When does Merten's product theorem accurately estimate the number of coprimes in an interval?
Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:
Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For ...
- 965
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
6
votes
1
answer
576
views
Other implications of Zhang's method
I have been reading a bit about Zhang's proof and the associated Polymath8 project.
Though Tao's high level summary
http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-...
- 2,810
6
votes
2
answers
837
views
A generalization of strong primes
In this post we denote the sequence of prime numbers as $p_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature.
Definition. We define the $\theta$-strong primes, ...
6
votes
1
answer
172
views
Is the set of all solutions $x > 0$ to $ \pi(x) = \operatorname{li}(x)$ unbounded?
Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$?
Here, $\pi(x)$ denotes the ...
- 4,985
6
votes
1
answer
233
views
Is there a connection between the average 'compositeness' of a rational number and $\phi$ (golden ratio)?
Let $n\in N$, where $n = p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{m}^{k_{m}}$ for $p_{i}$ prime.
Define the 'density' of $n$ as:
$d(n) = \frac{(p_{1}+1)^{k_{1}}(p_{2}+1)^{k_{2}}...(p_{m}+1)^{k_{m}}}{n}$
...
- 375
6
votes
1
answer
826
views
Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$
I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...
- 1,049
6
votes
4
answers
845
views
Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$
I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...
user40023
6
votes
1
answer
367
views
Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive ...
- 71
6
votes
1
answer
653
views
On permuted sum of squares of primes in a list
We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, $\sigma_1(\cdot)$,$\sigma_2(\cdot),\dots,\sigma_t(\...
- 13.9k
6
votes
1
answer
499
views
Understanding Sylvester' s $1871$ paper of primes in arithmetic progression of the forms $4n+3$ and $6n+5$
The following is the proof of infinitude of primes in arithmetic progression of the form $4n+3$ and $ 6n+5$ done by Sylvester in $1871$ in his paper "On the theorem that an arithmetical progression ...
- 233
6
votes
1
answer
328
views
Reference Request for a result on divisors of $p-1$
I have seen this result in several places without an English reference:
There exist infinitely many primes $p$ such that $p-1=2q_1q_2$ where $q_1$ and $q_2$ are prime numbers with $q_1,q_2>p^{1/4}$...
- 537
6
votes
1
answer
246
views
The L-function of Q(-1/2) and the "number of prime $p\equiv 3$ divisors" function
In the framework of classical motives, there is no such thing as a motive $\mathbb Q(-\tfrac 12)$, i.e. a tensor root of $\mathbb Q(-1)$. There is one, however, in a more general setting of "...
- 4,015
6
votes
1
answer
481
views
Probabilistic Proofs of Key Number-Theoretic Results
Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$.
Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where ...
- 329
6
votes
1
answer
350
views
Counting smooth numbers in short intervals
I am reading a few papers about counting smooth numbers in the interval $[x, x+\sqrt{x}]$, including the work of Harman, and Matomaki.
Both authors mentioned that the Dirichlet polynomial techniques ...
- 211
6
votes
2
answers
319
views
Evolution of partial sum of a sequence of induced Dirichlet characters
Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.
Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...
- 1,199
6
votes
0
answers
381
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
6
votes
0
answers
230
views
A bias for runs in Legendre symbols?
$\newcommand\Legendre[2]{\genfrac(){}{}{#1}{#2}}$An odd prime $p$ defines the sequence $\Legendre1 p,\Legendre2 p,\dotsc,\Legendre{p-1}p$
of values of the Legendre symbol describing the quadratic ...
- 17.9k
6
votes
0
answers
465
views
On improvements of the GPY sieve
When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows:
$$
S(x)=\sum_{x&...
- 1,315
6
votes
0
answers
149
views
Dickson's conjecture for Beatty sequences
A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...
- 1,990
6
votes
0
answers
654
views
Generalized prime number theorem and Riemann Hypothesis for non-number math objects
My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
- 3,098
6
votes
0
answers
201
views
Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
- 1,826
6
votes
0
answers
435
views
Average value of $\prod_{p|d}{p-1\over p-2}$ for $d=nq$, $n\in{\mathbb N}$, with $p$ odd prime
$\newcommand{\mean}{\mathop{\mathrm{mean}}}$
Define
$$
S(d) = \prod_{p|d\atop p>2}{p-1\over p-2}.
$$
Bombieri and Davenport (1966) proved that
$$
\mean\limits_{d\in{\mathbb N}} S(d) =
\mean\...
- 345
6
votes
0
answers
211
views
some problems on sum of two squares
During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....
- 841
6
votes
0
answers
257
views
Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$
For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$
with $T_1=1$, where $p_k$ denotes the $k$-th prime.
So multiplying by $(-1)^n$ and telescoping gives that for ...
6
votes
0
answers
206
views
Primes $p\in(n,2n)$ with $(\frac{-n}p)=-1$
Bertrand's postulate proved by Chebyshev states that for any $x>1$ there is a prime $p$ in the interval $(x,2x)$. In 2012 I considered some refinements of this by imposing additional requirement ...
- 15.6k
6
votes
0
answers
255
views
Gaussian square-free moat
Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free
Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$
For the analogous problem with Gaussian primes instead, ...
- 11.3k
6
votes
0
answers
233
views
admissible tuples vs. prime tuples
Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) ...
- 105k
6
votes
0
answers
505
views
$x^2+1$ attaining almost prime values
Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
- 3,062
6
votes
1
answer
1k
views
Arguments for the second Hardy–Littlewood conjecture being false?
Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that
$$\pi(x + y) - \pi(y) \leq \pi(x).$$
We can easily justify this heuristically, since
$$
\textrm{...
- 965
5
votes
3
answers
3k
views
Asymptotics of Product of consecutive primes
I am looking for the asymptotic growth of product of consecutive primes. Is there anything that is known about this growth?
- 13.9k
5
votes
2
answers
794
views
Is the result of Schmidt conditional to RH
From this page:
https://en.wikipedia.org/wiki/Chebyshev_function#Asymptotics_and_bounds
A theorem due to Erhard Schmidt states that, for some explicit positive constant $K$, there are infinitely ...
- 1,197
5
votes
1
answer
750
views
Geometric mean of prime factors of all numbers up to n
Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \...
- 1,228
5
votes
2
answers
1k
views
A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis
Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
- 1,018
5
votes
3
answers
2k
views
Goldbach conjecture and other problems in additive combinatorics
The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance:
$S = T$ is the set ...
- 3,098
5
votes
2
answers
1k
views
Error term in Mertens' third theorem
Mertens' third theorem states that:
$$\prod_{\substack{
p \leq x \\
\text{p prime}
}} \left( 1 - \dfrac{1}{p} \right) \sim \dfrac{e^{-\gamma}}{\log(x)}$$
Question: what is the best functions (...
- 521
5
votes
2
answers
435
views
Proving certain inequality related to Primes
I was reading the following paper. But I can't understand why the last line concerning $\frac{2}{\pi}$ is true. The proof is a work of Sylvester.
I would be happy if someone helps me in understanding ...
- 233
5
votes
1
answer
811
views
A consequence of Firoozbakht's conjecture?
This is a question out of curiosity, while looking at the Firoozbakht's conjecture. It might not be research related, but as usual, I am not really sure if a question ever is research related or not, ...
- 3,064
5
votes
1
answer
737
views
Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
user344548
5
votes
3
answers
809
views
Positive proportion of logarithmic gaps between consecutive primes
For $x, \lambda > 0$, define
$$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$
where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the ...
- 113
5
votes
1
answer
611
views
Why does this convolution of the prime counting function $\pi$ look like a parabola?
In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The ...
- 3,064
5
votes
1
answer
458
views
Moments of merit
The merit of a prime gap equals $(p_{n+1}-p_n)/\ln p_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very ...
- 17.6k
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 ...
user95470
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 ...
- 5,470