All Questions
Tagged with dirichlet-series nt.number-theory
33 questions with no upvoted or accepted answers
11
votes
0
answers
530
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 $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
- 2,480
9
votes
0
answers
546
views
Modern treatment of Delange's Tauberian Theorem
Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper:
H. Delange - Généralisation du théorème de ...
- 21.3k
7
votes
0
answers
146
views
Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?
The Selberg class of $L$-functions are Dirichlet series
$$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$
satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
- 1,570
6
votes
0
answers
286
views
Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum
Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
- 20.2k
6
votes
0
answers
233
views
Mean value theorem for Dirichlet series of prime support?
Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973),
$$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$
...
- 20.2k
5
votes
0
answers
95
views
Possible extension of Ikehara's theorem for Dirichlet series with not necessarily positive coefficients?
I'm wondering about a possible extension of Ikehara's theorem for Dirichlet series with coefficients that are not necessarily positive. Consider the following:
Let $D(s) = \sum_{n \geq 1} \frac{a(n)}{...
- 611
5
votes
0
answers
326
views
Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
\frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
- 1,199
4
votes
0
answers
265
views
Ramanujan-like series for $1/\pi^m$ and Dirichlet L-values
A rational Ramanujan-like series for $\pi^{-m}$ and character $\chi$ is a series with rational parameters which is of the following form:
$$ \sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m}
\frac{(s_i
)_{n}...
- 1,389
4
votes
0
answers
216
views
Gap Between Abscissae of Conditional Convergence and Holomorphicity for Dirichlet Series
For a Dirichlet series, $D = \sum_n a_n n^{-s}$ we may define the abscissae, in (non-strictly) increasing order
$\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, ...
- 576
4
votes
0
answers
259
views
multiple zeros of an L-function
I once heard a conjecture that a primitive L-function does not have multiple zeros except the central point of the critical strip.
Question:Why it is reasonable to conjecture a primitive L-function ...
- 3,337
3
votes
0
answers
139
views
Do the denominators of A006571(n)/A366450(n) have a Dirichlet generating function? Because they partially match A071974(n) and A056622(n)?
Consider the expansion: $$A006571(n) = q \prod _{k=1}^{\infty } \left(1-q^k\right)^2 \left(1-q^{11 k}\right)^2 \label{1}\tag{1}$$
A006571
and the triple sum:
$$A366450(n)=\sum _{k=1}^n \left(\sum _{y=...
- 1,183
3
votes
0
answers
167
views
A sharper estimate for a generalization of the sum-of-divisors function
I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$
This ...
- 31
3
votes
0
answers
79
views
Continuation of $\sum \sigma_\nu(n) a(n) n^{-s}$ for $a(\cdot)$ coming from a half-integral weight form
In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...
- 1,570
3
votes
0
answers
97
views
Supremum of certain modified zeta functions at 1
Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by
$$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$
where $(D/m)$ denotes the Jacobi symbol. ...
3
votes
0
answers
139
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
$$
\...
- 2,207
3
votes
0
answers
192
views
Determining coefficients of a Dirichlet series based on values on a vertical line
Let us suppose we have a Dirichlet series
$$ D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$
and that we know the values of $D(\tfrac{1}{2} + im)$ for $m \in \mathbb{Z}$. Can we recover the coefficients $a(...
- 1,570
2
votes
0
answers
131
views
Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n$...
- 3,098
2
votes
0
answers
188
views
How to best approximate $1/\zeta(s)$ by a finite sum
I would like to approximate $1/\zeta(s)$ for $s=1+it$ by a finite sum:
$$\frac{1}{\zeta(1+it)} = \sum_n \frac{\mu(n)}{n} \eta\left(\frac{n}{x}\right) +
\epsilon(t)$$
with $\eta$ a function of compact ...
- 20.2k
2
votes
0
answers
135
views
Question regarding proof of Mertens' estimates in Montgomery-Vaughan's "Multiplicative number theory"
I have been trying to read Theorem 2.7 of Montgomery-Vaughan's "Multiplicative number theory" volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write
$$\...
- 81
2
votes
0
answers
199
views
Multiplicity of zeros of partial sums of the Dirichlet Eta function
I am studying ways to approach the problem of the multiplicity of zeros of the partial sums of the Dirichlet Eta functions:
$$
\sum_{n=1}^{K}\frac{(-1)^{n-1}}{n^{s_o}} = 0
$$
more in particular, ...
- 362
2
votes
0
answers
341
views
Dirichlet series of powers of the prime omega function
Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$.
I was wondering what is known about the dirichlet series
$$\sum_{n=1}^{\infty}\frac{\omega(n)^k}{n^s},$$
in ...
- 195
2
votes
0
answers
451
views
Analytic continuation of "composite" zeta function
Let us define the Dirichlet series $$\mathcal C(s):=\sum_{n\text{ composite}}\frac{1}{n^s},\quad P(s):=\sum_{p\text{ prime}}\frac{1}{p^s}.$$
They are absolutely convergent in the half-plane $\sigma>...
- 21
2
votes
0
answers
425
views
Analytically continuing the limit of this series?
Main Question
I believe the following formula gives the right answer:
$$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times \sum_{r=1}^\...
- 89
1
vote
0
answers
120
views
Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$
This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted.
Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
- 1,126
1
vote
0
answers
112
views
If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?
Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
- 3,098
1
vote
0
answers
381
views
Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$
I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
- 3,098
1
vote
0
answers
166
views
Euler product over subsets of primes
It is well known that
$$\prod_p\,(1-p^{-1})=\frac 1 {\zeta(1)}=0$$
Given an arbitrary prime $\,q\,$ is it true that
$$\prod_{q\,|\,p+1}\,(1-p^{-1})=0\;\;\;?$$
Thanks.
- 1,008
1
vote
0
answers
102
views
Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune
In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
1
vote
0
answers
325
views
Extension of Heath-Brown's fourth moment $\sum_ \chi \lvert L (s, \chi) \rvert^4$ to complex characters
Let $L(s, \chi)$ denote the Dirichlet $L$-function associated to the character $\chi$.
In his paper A Mean Value Estimate for Real Character Sums, Heath-Brown proves a mean value bound for $L(s,\chi)$...
- 1,570
1
vote
0
answers
182
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 ...
- 1,199
1
vote
0
answers
98
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 ...
- 2,480
0
votes
0
answers
78
views
Could the patterns in the roots of the Riemann-Liouville differintegral of $(s-1)\,L(s, \chi_{q,j})$ be explained?
This question aims to extend this question to (automorphic) Dirichlet L-functions.
Take: $$f(s,q,j) = (s-1)\,L(s, \chi_{q,j})$$
with $q$ the modulus and $j$ the index of a character $\chi$. A fast way ...
- 4,169
0
votes
0
answers
151
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 $d_z(n)$ ...
- 627