All Questions
Tagged with dirichlet-series nt.number-theory
11 questions
14
votes
1
answer
571
views
Mean value theorem for Dirichlet series - optimize?
Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality
$$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq
\sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$
...
- 20.2k
12
votes
1
answer
663
views
Error term when truncating series for $1/\zeta(s)$
Let $s=\sigma+it$, $0\leq \sigma\leq 1$, $|t|\geq 1$, say. Using Euler-Maclaurin, one can easily show that, for $x\geq |t|$,
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
- 20.2k
8
votes
1
answer
2k
views
Dirichlet's theorem for number fields
I'd like to see a formulation of Dirichlet's theorem for number fields, i.e. some analogue of the assertion:
The number of primes less than $N$ congruent to $a \pmod{m}$ where $(a,m)=1$ is
$\frac{...
- 543
6
votes
0
answers
233
views
Mean value theorem for Dirichlet series of prime support?
Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973),
$$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$
...
- 20.2k
5
votes
1
answer
1k
views
Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?
This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...
- 4,169
5
votes
0
answers
326
views
Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
\frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
- 1,199
3
votes
1
answer
756
views
Special values of Hecke L-function
The Dedekind zeta function for a number field $K$ is defined as
$\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$.
By attaching a Hecke character $\psi$, we can define $L(s,\psi)=\sum_{I\...
- 453
3
votes
1
answer
846
views
Multiplicative functions whose Dirichlet series have essential singularities
What can be said about the partial sums of a complex-valued completely multiplicative function, let's say bounded by 1 in absolute value, if its Dirichlet series has an essential singularity?
As a ...
- 1,671
2
votes
1
answer
1k
views
On the convergence of Dirichlet series over the Mobius Mu function
It is known that if $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 1/2$ then RH holds. My question is:
Under RH why is it not $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = ...
- 731
2
votes
0
answers
188
views
How to best approximate $1/\zeta(s)$ by a finite sum
I would like to approximate $1/\zeta(s)$ for $s=1+it$ by a finite sum:
$$\frac{1}{\zeta(1+it)} = \sum_n \frac{\mu(n)}{n} \eta\left(\frac{n}{x}\right) +
\epsilon(t)$$
with $\eta$ a function of compact ...
- 20.2k
0
votes
2
answers
392
views
What is the relationship between the abscissa of holomorphy and abscissa of convergence of a Dirichlet series
Given a Dirichlet series $$\phi(s)=\sum_{n\ge1}\frac{a_n}{n^s}$$
let $\sigma_{\text{conv}}\in\bar{\mathbb{R}}$ its abscissa of convergence, then we know that $\phi(s)$ is holomorphic on the half-plan $...
- 93