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3
votes
0answers
63 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 $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
...
1
vote
1answer
130 views
Is there a Poisson Summation formula for imprimitive Dirichlet characters?
I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?
For a primitive Dirichlet character $\chi$ we have:
...
10
votes
2answers
217 views
Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that ...
1
vote
1answer
161 views
Abscissa of absolute convergence of the product of two Dirichlet series
I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
0
votes
0answers
69 views
Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?
If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as
$$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$
where $d_z(n)$ ...
2
votes
1answer
156 views
Special values of Hecke L-function
The Dedekind zeta function for a number field $K$ is defined as
$\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$.
By attaching a Hecke character $\psi$, we can define ...
1
vote
0answers
79 views
Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?
Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely ...
4
votes
1answer
190 views
A lower bound on the $L^2$ norm of a Dirichlet polynomial
The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below
\begin{equation*}
\int_0^{T} \Big| \sum_{\alpha T ...
6
votes
0answers
196 views
Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$
If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log ...
7
votes
0answers
166 views
If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?
One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
2
votes
1answer
318 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} ...
5
votes
3answers
226 views
von Staudt-Clausen for other special values
The von Staudt-Clausen theorem expresses that the Bernoulli numbers' denominators have a very special form (see the wikipedia page on the theorem for more details).
What interests me is that those ...
0
votes
1answer
156 views
The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros
My current question is concerned with a reference (paper or book) containing a proof of this result: The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros.
7
votes
2answers
338 views
Recovering $\sum_{n \leq x} a(n)$ from $\sum_{n \leq x} a(n)e^{-n/x}$
In the theory of automorphic forms and multiple Dirichlet series, we often take inverse Mellin transforms of Dirichlet series to come up with Tauberian theorems, like the Ikehara Tauberian method. In ...
8
votes
0answers
684 views
Analytic continuation of the Dirichlet generating series of the multiplicative partition function
Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series:
...
0
votes
2answers
160 views
A rapidly-converging series of the Hasse–Weil L-function associated with an elliptic curve over rationals
I know that for some L-series there is still a rapidly-converging series. My question is about the existence of a such a series for the Dirichlet series of the Hasse–Weil L-function associated with an ...
4
votes
1answer
426 views
An application of Mobius Inversion in a paper of Shintani
I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On ...
3
votes
0answers
232 views
multiple zeros of an L-function
I once heard a conjecture that a primitive L-function does not have multiple zeros except the central point of the critical strip.
Question:Why it is reasonable to conjecture a primitive L-function ...
6
votes
2answers
815 views
Divergence of Dirichlet series
Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
I asked ...
3
votes
1answer
583 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function.
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
2
votes
1answer
427 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 ...
7
votes
4answers
1k views
Introduction to L-series and Dirichlet characters?
I'm looking for an introductory text on Dirichlet characters and the L-series of a field K, specifically for quartic extensions of $\mathbb{Q}$. I have Davenport's Multiplicative Number Theory, ...
0
votes
1answer
160 views
Truncated Dirichlet series take their supremum on the imaginary axis
Hi there,
I am struggling with a theorem about truncated Dirichlet series. I am trying to prove the following theorem:
Let $(a_n)_n \subset \mathbb{C}$ and $N \in \mathbb{N}$. Then $\sup_{t \in ...
1
vote
1answer
435 views
Dirichlet Series Question
Consider $a_n$ a real valued sequence and define $D_{1,1,1}(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$ which converges in some half plane $\Re s =c.$ Define $D_{r,h,k}(s) = \sum_{n=1}^\infty \frac{e^{2\pi ...
4
votes
2answers
755 views
Some Dirichlet series questions.
I asked this question on m.SE in an attempt to find out the right words to say for these questions I am about to ask.
In his great answer, Matthew Emerton explained that (cuspidal) automorphic ...
7
votes
1answer
769 views
Dirichlet series expansion of an analytic function
Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
...
3
votes
1answer
252 views
Distribution of associate primes modulo q in number fields
In Dirichlet's theorem for number fields, I asked about an analogue of Dirichlet's theorem (or I guess I should call it the Prime Number Theorem for Arithmetic Progressions) for number fields. ...
5
votes
1answer
904 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
...
4
votes
1answer
328 views
A plausible positivity
After getting stuck with the
previous positivity
(it probably sounds too complex),
I would like to give a version of the problem which is of most interest to me.
Consider a sequence of real numbers
...
0
votes
1answer
412 views
Analytical continuation of a Dirichlet series with periodic coefficients
Fix a complex number s and a real number x, does there exist an analytic continuation of the Dirichlet series
$L(s,x):=\sum_{k=1}^{\infty}\frac{\sin^2(2\pi k x)}{k^s}$
to the whole complex plane ...
7
votes
3answers
1k views
Convergence of L-series
I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function
satisfies the ...
5
votes
1answer
296 views
Convergence of a sequence of continuable Dirichlet series
Let's say $f$ is a Dirichlet series which converges on the half-plane $\text{Re }s>\sigma$ to a function $f(s)$. Suppose further that $f(s)$ admits an analytic continuation to an entire function, ...
8
votes
2answers
881 views
Analytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1
A sequence $a_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a_{mn} = a_m a_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum_n a_n n^{-s}$, absolutely convergent for $\Re s > 1$. If in ...
2
votes
1answer
358 views
Dirichlet L series and integrals
If $f : t \to e^{-xt}$ with $x \geqslant 1$, and $d_n$ is the number of positive integers that divide $n$, I can show that
$$ \lim_{\epsilon \to 0^+} \sum_{n\geqslant 1}\frac{\epsilon^2 d_n ...
2
votes
2answers
372 views
Continuation up to zero of a Dirichlet series with bounded coefficients
Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a ...
1
vote
1answer
205 views
Shifted Dirichlet series
If $\sum_{n=1}^\infty \frac{a_n}{n^s} $ converges, does
$\sum_{n=1}^\infty \frac{a_n}{(n+1)^s} $ also converge?
2
votes
3answers
495 views
Dirichlet series whose coefficients are the bits of sqrt(2)
$$F\\,(s) = 1/1^s + 0/2^s + 1/3^s + 1/4^s + 0/5^s + 1/6^s + 0/7^s + 1/8^s + ... $$
The coefficients of this Dirichlet series are the base-2 digits of $\sqrt2$.
Does it have an analytic continuation ...
6
votes
4answers
483 views
multi-index Dirichlet series
Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form ...
7
votes
3answers
778 views
Is the maximum domain to which a Dirichlet series can be continued always a halfplane?
Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in ...
