The dirichlet-series tag has no usage guidance.

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308 views

### Analytically continuing the limit of this series?

Main Question
I believe to following formula gives the right answer:
$$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times ...

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44 views

### Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694
Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...

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74 views

### References for “quadratic” Dirichlet series

(Please pardon the use of nonstandard terminology, as I know not the accepted names for my entities of interest.)
Some personal research I have been doing led me to consider series of the form
...

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votes

**1**answer

110 views

### Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...

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69 views

### Square integral of finite Euler product

Consider the finite Euler product
$$
P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right).
$$
(Here $p_1, p_2, \dots$ are of course the primes.)
Question: What is a good asymptotic upper bound for
$$
...

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**1**answer

220 views

### How do I evaluate this sum for $s$ is a complex variable :$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$?

This question related to this question in SE ,I would like to know how do I
evaluate this sum for $s$ is a complex variable :$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$$ .
Edit01:And I think ...

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91 views

### An apparent closed form for a slightly tweaked Dirichlet L-function. Could it be proven? [closed]

I made a small tweak to the well-known Dirichlet L-function ($p$=prime):
$$L(s, \chi_4) :=\prod_p \bigg(\frac {p^s}{p^s-\chi_4(p)} \bigg)=\prod_p \bigg(\frac {p^s}{p^s-\sin\left(\frac{p ...

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**2**answers

357 views

### Is there a nice generating function proof of the following identity?

Consider the Jordan function $J_2(n)$ defined by
$$
J_2(n) = \#\{x \in (\mathbb{Z}/n)^2 \mid ord(x) = n\}
$$
(this is OEIS A007434). One can prove the following identity pretty easily:
$$
\sum_{d \mid ...

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78 views

### Determining coefficients of a Dirichlet series based on values on a vertical line

Let us suppose we have a Dirichlet series
$$ D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$
and that we know the values of $D(\tfrac{1}{2} + im)$ for $m \in \mathbb{Z}$. Can we recover the coefficients ...

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**1**answer

186 views

### Does $\prod_{n=2}^{\infty} \left(\frac {1}{1-\frac{\chi_k(n)}{n^s}} \right)$ converge for non-principal characters for all $\Re(s) > \frac12$?

This question loosely builds on this one.
Take the following infinite product:
$$N(s,\chi_k)=\prod_{n=2}^{\infty} \left(\frac {1}{1-\dfrac{\chi_k(n)}{n^s}} \right)$$
with $\chi_k$ a Dirichlet ...

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**1**answer

326 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:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...

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181 views

### Smoothing Dirichlet Series partial sums by means of Pontifex Path Bending Functions

I have recently stumbled upon a curious approach to smoothing of the partial sums of Ordinary Dirichlet Series, and which I would like to share with others. Theorem 11.18 of Apostol's "Introduction to ...

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61 views

### Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance:
$y[n] = x[n] + y[n-1]$
$Y(z) = X(z) + Y(z) ...

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87 views

### Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?)
But I would like to know if ...

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**1**answer

131 views

### Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like :
$$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...

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70 views

### Extracting information from $\sum_{n \leq X} a(n) (X-n)^d$

Allow me to give context to the question, which appears in the box at the bottom.
A very general hope from the theory of Dirichlet series is to try to extract information about the coefficients of a ...

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**1**answer

469 views

### Completely multiplicative functions with values in $\{-1,1\}$

This question is from Eric Saias and myself:
Let $A$ be the set of abscissas of convergence of Dirichlet Series $\sum_{n\ge 1} \frac{f(n)}{n^s}$
where $f(n)$ is completely multiplicative and $f(n) \in ...

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**1**answer

432 views

### Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...

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**1**answer

102 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 ...

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194 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 $\Re(s)>\alpha$ ...

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102 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) =
...

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**1**answer

190 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:
...

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**2**answers

283 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 ...

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**1**answer

263 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 ...

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105 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 $d_z(n)$ ...

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**1**answer

225 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 ...

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89 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 ...

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**1**answer

233 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 ...

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239 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 ...

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220 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 ...

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**1**answer

473 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} ...

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**3**answers

302 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 ...

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**1**answer

164 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.

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385 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 ...

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**0**answers

754 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:
...

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**2**answers

168 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 ...

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**1**answer

499 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 ...

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234 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 ...

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**2**answers

914 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 ...

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**1**answer

688 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 ...

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votes

**1**answer

483 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 ...

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**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, ...

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**1**answer

170 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 ...

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**1**answer

460 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 ...

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**2**answers

808 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 ...

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**1**answer

984 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 $\forall \sigma >\sigma_a:$
...

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**1**answer

259 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. ...

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**1**answer

1k 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
...

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**1**answer

331 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
...

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**1**answer

425 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 ...