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Results tagged with prime-numbers
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user 101078
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
9
votes
Accepted
Covering the primes with pairs of consecutive integers
For large enough $n$ no such set of integers exist.
First of all, let us say that $a$ covers prime $p$ if $a$ or $a+1$ is divisible by $p$.
Now, if $p$ is a prime with $\frac{n+1}{2}<p\leq n$, $a\leq …
- 7,193
9
votes
Accepted
Condition for $8p+1$ divides $(2^p+1)/3$?
This follows easily from the octic reciprocity law for $2$. More precisely, it states that if $q$ is a prime of the form $q=8n+1$ and $n$ is odd, then $q=a^2+256b^2$ for some integers $a,b$ (note that …
- 7,193
1
vote
A special kind of multiplicative function $f: \mathbb N \to \mathbb N$ such that $f(p)=p+k$ ...
Suppose that the Schinzel's hypothesis is true. It suffices to assume that for any pair of coprime integers $a$ and $b$ with $a>0$ and such that $(a,b) \neq (c^2,-d^2)$ for any integers $c$ and $d$ th …
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6
votes
Accepted
Estimating a sum over prime numbers
Using the fact that
$$\sum_{y\leq p\leq 2y} \frac{\log p}{p}\ll 1,$$
we get for any $A\geq 2$
$$\sum_{x/2A\leq p\leq x/A} \frac{\log p}{p\log(x/p)}\ll \frac{1}{\log A}$$
Hence for any positive int …
- 7,193
19
votes
Accepted
Are there any papers about this observation of the distribution of the zeros of the zeta fun...
This is called Landau's formula. More precisely, if we extend the von Mangoldt function $\Lambda(n)$ to the function $\Lambda:\mathbb R_+\to \mathbb R$ by $\Lambda(x)=0$ for non-integer $x$, then
$$
\ …
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6
votes
Counting squares modulo $p$ that are also prime in an interval
Here is the paper by P. Pollack on the distribution of non-residues and residues. Theorem 1.3 states that for any $\varepsilon>0$, $A<\infty$ and large enough $m$ there are at least $(\ln m)^A$ prime …
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8
votes
Accepted
Is it true that $\det\big[\sin 2\pi\frac{(j-k)^2}p\big]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}...
We will use the notation $e_p(t)=\exp\left(\frac{2\pi it}{p}\right)$. First, let us show that for any $1\leq m\leq \frac{p-1}{2}$ there is an eigenvector of your matrix $A_p$ with eigenvalue
$$
\lambd …
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4
votes
Increasing sequences and Wieferich primes
I will focus on Q3. Let us evaluate $J(2^{2^n}+1)$. Denote $N=2^{2^n}+1$. Then $N-1=2^{2^n}$. Notice that $2^{2^{n+1}}\equiv 1 \pmod N$. Let $M=\frac{N-1}{2^{n+1}}=2^{2^n-n-1}$, which is an integer. W …
- 7,193
23
votes
Does the average primeness of natural numbers tend to zero?
The answer to Question 1 is "yes".
To see this, notice that $s_n$ is at most square root of the average square of divisor, i.e.
$$
s_n\leq \sqrt{\frac{\sum_{d\mid n}d^2}{\sum_{d\mid n} 1}}=\sqrt{\frac …
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4
votes
Accepted
Robin's inequality and the zeros of the Riemann zeta function
It is well-known that the Riemann hypothesis implies
$$
\theta(x)=x+O(\sqrt{x}\ln^2 x).
$$
Therefore, under the Riemann hypothesis we have
$$
\ln\theta(x)=\ln x+O\left(\frac{\ln^2 x}{\sqrt{x}}\right). …
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3
votes
Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?
Obviously, abscissa of convergence is at most 1, because the series is dominated by the series for $\zeta(s)$. In fact, it is equal to 1. Indeed, assume that your series converges for some $s=1-\varep …
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6
votes
Accepted
A generalization Bertrand's postulate
First of all, as mentioned by Random above, this is a very strong conjecture, because it is stronger than Legendre's conjecture. As far as I know, it is not known even if we assume the truth of Rieman …
- 7,193
7
votes
Accepted
Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions ...
This identity is false. To see that, notice first that
$$
\sum_{p\leq \sqrt{2n-3}}a_p(n)=2\sum_{p\leq \sqrt{2n-3}}1-\sum_{\substack{p\leq \sqrt{2n-3}\\ p\mid n}}1=2\mathrm{ord}_C(n)-O(\ln n),
$$
so
$$ …
- 7,193
12
votes
Accepted
A consequence of Firoozbakht's conjecture?
Notice that
$$
1\leq \frac{\log p_{n+1}}{\log p_n}=1+\frac{\log(p_{n+1}/p_n)}{\log p_n}\leq 1+\frac{p_{n+1}-p_n}{p_n\log p_n}.
$$
The last inequality is due to $\log(1+x)\leq x$ for $x\geq 0$.
Now, fo …
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1
vote
Asymptotic for a number theoretic sequence and its Dirichlet series' convergence
Your series is never absolutely convergent in any half-plane of the form $\mathrm{Re}\,s>\delta$ with $\delta<1$ and there is even no convergence in the case $\mathrm{Re}\,s=1$. To prove this, let us …
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