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0answers
67 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
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1answer
169 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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0answers
180 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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0answers
147 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
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1answer
289 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
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3answers
223 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
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1answer
150 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
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2answers
323 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
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0answers
667 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
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2answers
153 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
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1answer
409 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
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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 ...
5
votes
2answers
764 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
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1answer
567 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
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1answer
416 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
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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
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1answer
158 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
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1answer
430 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
744 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
734 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
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1answer
251 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
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1answer
850 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
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1answer
327 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
407 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
295 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
864 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
354 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
365 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
202 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?
1
vote
3answers
492 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 ...
5
votes
4answers
473 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
754 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 ...