Questions tagged [dirichlet-series]
The dirichlet-series tag has no usage guidance.
179
questions
17
votes
1
answer
687
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 ...
16
votes
1
answer
666
views
Dirichlet series with a single zero
I need to find a Dirichlet series f that has the following property.
f is zero in only one point s such that Re(s) > $\sigma_c $.
15
votes
2
answers
703
views
If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$?
Helmut Hasse has proved that for $s \in \mathbb{C}-\{1\}$ the Riemann zeta function can be written as:
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}...
15
votes
1
answer
2k
views
Dirichlet series expansion of an analytic function
Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}...
15
votes
3
answers
858
views
Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $
I am trying to prove or disprove
$$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$
where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
14
votes
3
answers
2k
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 ...
14
votes
1
answer
719
views
Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
14
votes
1
answer
552
views
Mean value theorem for Dirichlet series - optimize?
Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality
$$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq
\sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$
...
14
votes
0
answers
1k
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:
$$\kappa(s)=\prod_{m=2}^{\infty}\frac{1}{1-m^{-s}}=\sum_{n=1}^{\infty}\frac{\...
12
votes
3
answers
2k
views
Are L-functions uniquely determined by their values at negative integers?
Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that
the corresponding L-function
$$L_{\{a_n\}...
12
votes
2
answers
380
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 $...
12
votes
1
answer
567
views
Convergence of the series involving Mobius functions $\sum_{k,d} \mu(d) x_{kd}$
(I originally asked this question here, but the problem appears much more difficult than I think after a moment of thought, so I think it might be more suitable to post it here. Please tell me if this ...
12
votes
2
answers
520
views
Computation of modified Gauss sums
Let $\chi$ be a primitive Dirichlet character of conductor $q$. I want to compute numerically $$G(k)=\sum_{n\bmod q}\chi(n)e^{2\pi i n(n-k)/(2q)}$$
for all $k$ with $0\le k<2q$ with $k\equiv q\...
12
votes
1
answer
717
views
If the generating function summation and zeta regularized sum of a divergent series 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 ...
12
votes
1
answer
606
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 $x\geq |t|$,
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
12
votes
0
answers
526
views
Additive and multiplicative convolution deeply related in modular forms
From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
11
votes
3
answers
825
views
Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$
I was working with some Dirichlet series and I realized that I have never seen any general conditions under which
\begin{equation}
\sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
11
votes
0
answers
510
views
Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$
If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
9
votes
2
answers
2k
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 ...
9
votes
1
answer
664
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 in ...
9
votes
2
answers
1k
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 ...
9
votes
0
answers
505
views
Modern treatment of Delange's Tauberian Theorem
Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper:
H. Delange - Généralisation du théorème de ...
8
votes
4
answers
3k
views
Modern Algebraic Geometry and Analytic Number Theory
I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (...
8
votes
2
answers
1k
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 (...
8
votes
2
answers
529
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
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{...
8
votes
1
answer
388
views
Is $\frac{1}{L(1+it)}$ unbounded?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
8
votes
4
answers
650
views
multi-index Dirichlet series
Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s_1,...,s_k)=\sum_{(n_1,...,n_k)\in\mathbb{N}^k}\frac{a_{n_1,...,n_k}}{n_1^{s_1}......
8
votes
0
answers
104
views
What is known about the following series?
For $k\in{\mathbb Z}^2$ write $|k|=\sqrt{k_1^2+k_2^2}$ for the euclidean norm. Then let $g(k)=gcd(k_1,k_2)$.
For $s\in\mathbb C$ let
$$
D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
7
votes
4
answers
2k
views
Introduction to L-series and Dirichlet characters?
I'm looking for an introductory text on Dirichlet characters and the L-series of a field K, specifically for quartic extensions of $\mathbb{Q}$. I have Davenport's Multiplicative Number Theory, ...
7
votes
2
answers
850
views
Extracting Dirichlet series coefficients
Cauchy's integral formula is a powerful method to extract the $n$'th power series coefficient of an analytic function by evaluating a single complex integral. Is there any such analytic method to ...
7
votes
3
answers
817
views
von Staudt-Clausen for other special values
The von Staudt-Clausen theorem expresses that the Bernoulli numbers' denominators have a very special form (see the wikipedia page on the theorem for more details).
What interests me is that those ...
7
votes
2
answers
371
views
Proving a series for $\pi$ by Plouffe
Simon Plouffe found experimentally a series for $\pi$ that can be written as
$$\frac{\pi}{24} = \sum_{n=1}^\infty \frac{1}{n} \left( \frac{3}{e^{\pi n}-1} -\frac{4}{e^{2\pi n}-1} +\frac{1}{e^{4\pi n}-...
7
votes
0
answers
137
views
Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?
The Selberg class of $L$-functions are Dirichlet series
$$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$
satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
6
votes
4
answers
405
views
Reference for behavior of Artin $L$-functions at $\Re(s) = 1$
Would anyone know a reference that proves the basic facts about Artin $L$-functions at $\Re(s) = 1$? Namely, the non-vanishing and holomorphicity for non-trivial characters.
I assume this was done in ...
6
votes
1
answer
113
views
Finding the continued fraction for the "tails" of $\eta(3)$
I am interested in the continued fractions for the "tails" or "correction term" of the series sum of specific constants. For example, the Madhava's correction term for $\pi/4$:
$$
\...
6
votes
1
answer
333
views
The Dirichlet series of the harmonic numbers
I'm curious about the Dirichlet series $$F(s) = \sum_{n = 1}^\infty \frac{H_n}{n^s}$$
of the sequence $H_n = \sum_{k = 1}^n \frac{1}{k}$ of harmonic numbers. Its abscissa of convergence is $1$. ...
6
votes
1
answer
311
views
Counting smooth numbers in short intervals
I am reading a few papers about counting smooth numbers in the interval $[x, x+\sqrt{x}]$, including the work of Harman, and Matomaki.
Both authors mentioned that the Dirichlet polynomial techniques ...
6
votes
0
answers
270
views
Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum
Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
6
votes
0
answers
199
views
Mean value theorem for Dirichlet series of prime support?
Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973),
$$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$
...
6
votes
0
answers
619
views
Generalized prime number theorem and Riemann Hypothesis for non-number math objects
My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
5
votes
1
answer
666
views
An application of Mobius Inversion in a paper of Shintani
I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On zeta-...
5
votes
1
answer
972
views
Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?
This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...
5
votes
1
answer
422
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
$...
5
votes
1
answer
460
views
If a Dirichlet series converges Conditionally, how can I apply Euler product?
In 1737, Euler discovered that if $ f(n) $ is multiplicative and $ \sum f(n)/n^{s} $ converges absolutely for ${\rm Re}(s) > \sigma_a$ then we have
\begin{equation}
\sum_{n=1}^{\infty} \frac{f(n)}{...
5
votes
2
answers
761
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:
...
5
votes
1
answer
166
views
Frequency of large values of the Mertens function
It is known that with $M(x) = \sum_{n\le x}\mu(n)$, there are infinitely many $x$ s.t. $|M(x)|\ge x^{\frac{1}{2} - \varepsilon}$ (see Chapter 15 of Montgomery-Vaughan, for example). Is there any way ...
5
votes
1
answer
198
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) ...
5
votes
1
answer
377
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, ...
5
votes
0
answers
153
views
Dirichlet eta function and Stirling Permutations
The Stirling permutations of order $k$ is a permutation of the multiset $1, 1, 2, 2, ..., k, k$. The Dirichlet $\eta$-function is a function closely related to the Riemann $\zeta$-function.
According ...