All Questions
Tagged with analytic-number-theory elementary-number-theory
383
questions
5
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1
answer
82
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Given nonzero $p(x)\in\mathbb Z[x]$. Are there infinitely many integers $n$ such that $p(n)\mid n!$ is satisfied?
This problem comes from a famous exercise in elementary number theory:
Prove that there are infinitely many $n\in\mathbb Z_+$ such that $n^2+1\mid n!$.
I know a lot of ways to do this. A fairly easy ...
- 786
2
votes
1
answer
53
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The "Euler Product formula" for general multiplicative functions
For the totient function $\phi$, we have the well known "Euler's product formula" (as named on Wikipedia) $$\phi(n) = n \prod_{p | n} \left( 1 - \frac{1}{p} \right)$$
This is easy to show ...
1
vote
1
answer
55
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$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?
I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function):
$$ R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}). $$
Over the interval $x=2$ to $x=10^4$ the average ...
- 754
1
vote
0
answers
32
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How do you parameterize simultaneous solutions to equations with expressions like "$ x +2 \left\lfloor\frac{x}{3}\right\rfloor + 1 - [3 \mid x]$"?
Let all functions be integer functions herein. I.e. $\Bbb{Z}\to\Bbb{Z}$ or $\Bbb{N}\to\Bbb{Z}$ where appropriate.
I found this jewel of floor functions.
So that made me wonder whether, we can solve ...
1
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1
answer
87
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The sum $f(x)= \sum_{p \le x} e^{\frac{1}{p\log p}}$ is very close to $\pi(x)$. Why is that?
The prime counting function $\pi(x)$ counts the number of primes less than a given $x$. There are other counting functions like Chebyshev's functions which count sums of logarithms of primes up to a ...
- 754
4
votes
0
answers
127
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If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$
Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
- 928
1
vote
1
answer
51
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Proposition 16.5.4 in Ireland-Rosen
We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over ...
- 4,429
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2
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70
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Exercise 1 Section 9.2. Montgomery/Vaughan's Multiplicative NT
I am trying to show that for any integer $a$, $$e(a/q) =
\sum_{d|q, d|a} \dfrac{1}{ϕ(q/d)} \sum_{χ \ (mod \ q/d)} χ(a/d) τ(χ).$$ First I considered the case $(a,q)=1$ and the mentioned equality holds. ...
- 281
2
votes
1
answer
84
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Proving Euler product related to Riemann zeta function
Let $\omega(n)$ denote the number of prime factors of a positive integer $n$. Prove that \begin{equation}\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}\end{equation}
My ...
user1289888
2
votes
1
answer
52
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Number of primitive Dirichlet characters of certain order and of bounded conductor
Writing $q(\chi)$ for the conductor of a Dirichlet character $\chi$, one can show using Mobius inversion that
$$\#\{\text{$\chi$ primitive Dirichlet characters}\,:\,q(\chi)\leq Q\}\sim cQ^2.$$
My ...
user1296122
0
votes
0
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20
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Proving a Dirichlet series relating to the zeta function [duplicate]
Prove that \begin{equation}\sum_{n=1}^{\infty}\frac{d(n^2)}{n^s}=\frac{\zeta(s)^3}{\zeta(2s)},\end{equation}where $d(n)$ denotes the number of divisors of $n$.
My solution: Observing that we have a ...
user1289888
1
vote
1
answer
84
views
Dirichlet series and Euler product
For a multiplicative function $f$, show that we have \begin{equation}\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).\end{equation}
My ...
- 135
-1
votes
1
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79
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What is known about the equation $x^2+ay^2=b^2$, where $a$ is a fixed square free positive integer and $b$ is a fixed positive integer. [closed]
$(b,0)$ and $(-b,0)$ are two trivial solutions. What do we know about the nontrivial solutions of the equation $x^2+ay^2=b^2$.
- 396
3
votes
1
answer
547
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A heuristic approach to the Prime number theorem
Let us consider the Sieve of Eratosthenes and the rough probability $(1-1/p)$ that a natural number $n \in \mathbb{N}$ is not divisible by a prime number $p<n$. If we assume that the conditions of ...
- 2,040
2
votes
0
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63
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Questions regarding the function $\Omega(n)$
For any positive integer $n$, define $\Omega(n)$ to be the number of prime factors (including repeated factors, so for example $\Omega(12)=\Omega(2^2\times 3)=3$). It is well known (Pillai-Selberg) ...
user1232857