All Questions
Tagged with dirichlet-series euler-product
20
questions
6
votes
0
answers
194
views
Proof of Theorem 1.1 of Analytic Number Theory by Iwaniec & Kowalski
I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski.
They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\...
- 1,145
1
vote
1
answer
78
views
How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.
Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
- 937
0
votes
1
answer
86
views
Does the Euler product for the Rankin-Selberg convolution of two eigenforms require the Ramanujan conjecture?
Suppose $f$ and $g$ are weight $k$ eigenforms for the Hecke operators, normalized, with Fourier coefficients $a_{n}$ and $b_{n}$ respectively. I wanted to see if I could derive a Euler product for
$$L(...
user427380
4
votes
1
answer
126
views
How to prove that $\sqrt{3}\pi/6=\prod_{p \equiv 1 \pmod{6}} \frac{p}{p-1}\prod_{p \equiv 5 \pmod{6}} \frac{p}{p+1}$ with $p \in \mathbb{P}$?
I would like to prove the formula $$\frac{\sqrt{3}\pi}{6}=\left(\prod_{\substack{p \equiv 1 \pmod{6} \\ p \in \mathbb{P}}} \frac{p}{p-1}\right) \cdot \left(\prod_{\substack{p \equiv 5 \pmod{6} \\ p \...
- 934
6
votes
1
answer
150
views
Abelian group zeta function
Let $s \in \mathbb{C}$. What's known about
$$\zeta_{\mathrm{ab}}(s) := \sum_G \frac{1}{o(G)^s} \tag{1}$$
where the sum is over all finite abelian groups $G$ up to isomorphism?
By the primary ...
- 7,631
5
votes
0
answers
327
views
Convergence of Euler product implies convergence of Dirichlet series?
(Crossposted to Math Overflow) Suppose we have an Euler product over the primes
$$F(s) = \prod_{p} \left( 1 - \frac{a_p}{p^s} \right)^{-1},$$
where each $a_p \in \mathbb{C}$. The Euler product is ...
- 5,830
5
votes
1
answer
248
views
Trouble Proving $\sum_{n=1}^\infty \frac{\mathrm{d}(n)^2}{n^s}=\frac{\zeta(s)^4}{\zeta(2s)}$
I am running into considerable trouble trying to prove the identity in the question. I figure the solution will come from Euler-products, so here was my attempt. I want to show that
$$
\sum_{n=1}^\...
- 1,968
6
votes
1
answer
304
views
Not-too-slow computation of Euler products / singular series
I'd like to compute, to at least a few digits of accuracy, the constants that arise in Hardy-Littlewood conjecture F / Bateman-Horn conjecture, in particular for just a single quadratic polynomial.
...
- 41.7k
0
votes
0
answers
48
views
Closed forms at integer values for Euler products with Dirichlet $\chi_{4}(p^s)$. Could these be extended towards non-integer values?
I was experimenting with the following products over all primes:
$$\prod_p \left(\frac{p^s}{{p^s-\sin\left(\dfrac{p^s \,\pi}{2}\right)}} \right)\,\cdot\,\prod_p \left(\frac{p^s+\sin\left(\dfrac{p^s \...
- 3,191
1
vote
1
answer
224
views
Euler product for $ \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 $
I was looking up a modular forms online: $S_3^{new}\big(\chi_8(3, \cdot)\big) $ it can be written as an Eta product:
$$f(z) = \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2
= q \prod_{n=1}^\infty (1 - q^n)...
- 24.5k
3
votes
0
answers
332
views
Convergence of Euler Product for Leibniz Pi Formula
In class we were shown a derivation of Leibniz's formula for pi:
$$\frac{\pi}{4}=\sum_{n=0}^\infty\frac{(-1)^n}{2n+1}$$
We can rewrite this formula using the following function on $\mathbb{N}$:
$$\...
- 3,405
1
vote
0
answers
79
views
Multiplicative coefficients of Selberg functions
Let $F(s)$ be a Selberg function, $F(s)=\sum_{n=1}^\infty\frac{a(n)}{n^{s}}$ and $\log F(s)=\sum_{n=2}^\infty\frac{b(n)\Lambda(n)}{\log n}\frac{1}{n^s}$ where $b(n)$ is zero unless $n$ is a positive ...
- 317
2
votes
2
answers
1k
views
Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)
On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula:
$\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$
where $s>1$, $\chi$ is a ...
- 1,055
1
vote
1
answer
110
views
Can be justified this formula for $\zeta(2n+1)$
Can be justified for integers $n\geq 1$ that
$$\zeta(2n+1)=\prod_{\text{p, prime}}\frac{1-\sigma(p^{2n})^{-1}}{1-p^{-2n}}?$$
Truly I don't know if I am wrong another time, when I use for an integer $...
user243301
7
votes
1
answer
749
views
Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?
The Dirichlet $\beta$-function is defined for $\Re(s)>0$ as:
$$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$
It has the following Euler product (I used that Dirichlet character $\chi_{4}(...
- 3,191