All Questions
Tagged with dirichlet-series elementary-number-theory
17
questions
4
votes
1
answer
47
views
How to find the $\zeta$ representation of a $L$-series
Consider the following problem:
Show that for $s>1$:
$$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.$$
($\mu$ denotes the Mobius function)
My approach:
One may first note that the ...
- 609
0
votes
0
answers
40
views
LCM sum with $\log $'s
If I want to evaluate $$\sum _{[r,r']\leq x}\log r\log r'$$ I could write it as an integral using Perron's formula, pick up a pole, and get a main term which involves looking at (the derivatives at $\...
- 1,662
0
votes
1
answer
530
views
Dirichlet series for $\zeta^3(s)/\zeta(2s)$.
I am currently studying number theory and our instructor refers to Apostol's book on Analytic number theory for the chapter Dirichlet series.In that book,there is an exercise which is as follows:
Let $...
- 3,619
4
votes
1
answer
223
views
Proof that Dirichlet series $\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2}$
So I want to prove the following:
$$\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2},$$
where $\omega(n)$ is the number of distinct prime factors of $n.$
I computed it to $10^{10}$ and it does ...
- 655
1
vote
0
answers
52
views
Variant of Möbius inversion: $b(n) = \sum_{d^2 \mid n} a(n/d^2) d^\alpha$
I'm trying to understand a step in a classic paper of Rankin. In Rankin's paper Contributions to the theory of Ramanujan's function $\tau(n)$ and similar arithmetical functions, he defines
$$ b(n) := \...
- 11
0
votes
1
answer
60
views
Why this group is cyclic?
Let $m$ be a positive integer, let $\chi$ be the Dirichlet character on $\mathbb{Z}/m\mathbb{Z}$ which means $\chi$ is a group homomorphism from $(\mathbb{Z}/m\mathbb{Z})^{*}$ to $\mathbb{C}^{*}$. We ...
- 915
3
votes
1
answer
226
views
Dirichlet's theorem/Bunyakovsky conjecture for infinite composites/a single prime
From https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, the Bunyakovsky Conjecture is an open problem that states that $f(x)$ has infinitely many primes in sequence $f(1),f(2),...$ if
1) The ...
- 1,950
0
votes
0
answers
83
views
Why is it impossible to invert the analytic continuation of a Dirichlet series?
By Mathematica (and the truncated Euler MacLaurin formula) I know that:
$$\zeta(s)=\lim_{k\to \infty } \, \left(\sum _{n=1}^k \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\right) \tag{1}$$
when the real part ...
- 7,448
1
vote
1
answer
56
views
Are there any minor extensions of Dirichlet's theorem?
For example, can we say that for $k$ $odd$, there are infinitely many primes of the form $a+bk$, for a fixed $a,b$ with $gcd(a,b)=1$?
How about for $k$ $odd$, there are infinitely many primes of the ...
- 1,950
1
vote
1
answer
122
views
Are there infinitely many primes of the form $k\cdot 2^n+1$ for a fixed $n$ and odd $k$
It is clear from Dirichlet's theorem on arithmetic progressions that for a fixed $n$, there are infinitely many primes of the form $k\cdot 2^n+1$ for a fixed $n$ and $k=1,2,3,..$. However, what if we ...
- 1,950
0
votes
1
answer
412
views
Question with Dirichlet convolution involving Mobius function and divisor function
So my question is:
Use the Dirichlet series to show that
$\sum_{k|n}\mu(k)d(\frac{n}{k})$ = 1
for all natural numbers n where d(.) is the divisor function.
I've just started learning about the ...
- 101
2
votes
0
answers
61
views
On Uniform Elementary Estimates of Arithmetic Sums Error Term
Stefan A. Burr's paper "On Uniform Elementary Estimates of Arithmetic Sums" has this result:
Suppose
$G(s)=\sum_{n=1}^{\infty}\frac{g(n)}{n^s},$ $G_2(s)=|g(1)|+\sum_{n=2}^{\infty}\frac{|g(n)...
- 55
1
vote
1
answer
685
views
Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$.
Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$.
I know that we should use absolute convergence but not sure how that applies in this case.
- 387
1
vote
1
answer
526
views
Is this Dirichlet series generating function of the von Mangoldt function matrix correct?
Let $\mu(n)$ be the Möbius function and let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
Let the matrix $T$ be defined as:
$$T(n,k)=a(...
- 7,448
4
votes
1
answer
516
views
Does the "alternating" harmonic series where only prime terms are negative converge?
We know that the harmonic series $\sum \frac{1}{n}$ diverges, yet the alternating harmonic series $\sum \frac{(-1)^n}{n}$ converges.
Euler famously gave a proof of the infinitude (and of the "density"...
- 4,204