The dirichlet-series tag has no wiki summary.

**0**

votes

**1**answer

89 views

### Dirichlet series without order term

is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions?
$D(s) = \sum_{0<n<N}a_n/n^s$
The ...

**4**

votes

**0**answers

151 views

### Equivalence of Euler products of Dirichlet series and Meromorphic continuation

Suppose $(f_n(s))_n$ and $(g_n(s))_n$ are two sequences of Dirichlet series with positive coefficients such that $\exists \alpha\in\mathbb R$ such that for all $s\in\mathbb C$ with $\Re(s)>\alpha$ ...

**3**

votes

**0**answers

73 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

**1**answer

134 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

**2**answers

227 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

**1**answer

181 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

**0**answers

77 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

**1**answer

167 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

**0**answers

83 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

**1**answer

196 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 ...

**7**

votes

**0**answers

200 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 ...

**8**

votes

**0**answers

176 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

**1**answer

337 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

**3**answers

229 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

**1**answer

157 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

**2**answers

343 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

**0**answers

692 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

**2**answers

161 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

**1**answer

441 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

**0**answers

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

**2**answers

825 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

**1**answer

589 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

**1**answer

437 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

**4**answers

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

**1**answer

161 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

**1**answer

441 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

**2**answers

765 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

**1**answer

791 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

**1**answer

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

**1**answer

919 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

**1**answer

329 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

**1**answer

414 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

**3**answers

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

**1**answer

297 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

**2**answers

892 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

**1**answer

362 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

**2**answers

377 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

**1**answer

208 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

**3**answers

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

**4**answers

488 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

**3**answers

784 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 ...