# Questions tagged [dirichlet-series]

For questions on Dirichlet series.

563 questions
Filter by
Sorted by
Tagged with
47 views

### How to find the $\zeta$ representation of a $L$-series

Consider the following problem: Show that for $s>1$: $$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.$$ ($\mu$ denotes the Mobius function) My approach: One may first note that the ...
• 609
101 views

### Zeta Lerch function. Proof of functional equation.

so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following. In the article "Note sur la function" by Mr. Mathias Lerch, a ...
2k views

### How does Wolfram Alpha know this closed form?

I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function: $$\zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3).$$ How does WA know ...
• 5,295
1 vote
32 views

### 2 concise tables of “usual” series (mostly trigonometrics) and of "usual" L-series (Zeta, Eta, Beta...)

CONTEXT Common series are usually described as infinite sums, written as consecutive terms ending with (…). Or they can be described using the $\sum_{}$ symbol and arguments usually including $(-1)^k$ ...
37 views

### Building the theoretical foundation for generating functions - formal power series

I have read several documents on generating functions. I would like to inquire about two issues: Among the materials I have read, some mention generating functions constructed from formal power ...
• 817
74 views

• 825
128 views

### Power series for $\sum_{n=0}^\infty(-1)^n/n!^s$ (around $s=0$)

I'm looking for ways to compute the coefficients of the power series $$\sum_{n=0}^\infty\frac{(-1)^n}{n!^s}=\sum_{k=0}^{\infty}c_k s^k$$ (a prior version of the question asked whether such an ...
• 40.1k
1 vote
75 views

### A question about Lemma 15.1 (Landau’s theorem for integrals) in Montgomery-Vaughan’s book

Lemma 15.1 in Montgomery-Vaughan’s analytic number theory book is Landau’s theorem for integrals. My question is, why is it necessary to have $A(x)$ bounded on every interval $[1,X]$? Doesn’t the ...
• 405
194 views

• 16.4k
35 views

### $0 = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)} \implies Re(s) \leq \frac{1}{2}$?

Define $f(s)$ as $$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)}$$ where we take the upper complex plane as everywhere analytic. Notice this is an antiderivative of the Riemann Zeta function, ...
• 16.4k
67 views

### Asymptotics for the number of $n\le x$ which can be written as the sum of two squares. Is Perron's formula applicable?

For all $n\ge 1$, let $$a_n = \begin{cases} 1\quad&\text{if n can be written as the sum of two squares;}\\ 0&\text{otherwise} \end{cases}$$ I am interested in $A(x):=\sum_{n\le x}a_n$. ...
• 8,421
279 views

### Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]

I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ...
1 vote
102 views

### Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$ for $\chi$ mod $5$

My HW question is: Evaluate the series $$L(1, \chi_5) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$$ where $\chi_5$ is the unique nontrivial Dirichlet character mod $5$. My work is: \begin{align*} ...
• 3,187
116 views

### Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n}$ for $\chi$ mod $3$

Here is the homework question I am working on: Evaluate (as a real number) the series $$L(1, \chi_3) = \sum_{n=1}^\infty \frac{\chi_3(n)}{n},$$ where $\chi_3$ is the unique nontrivial Dirichlet ...
• 3,187
273 views

### Dirichlet series with infinitely many zeros

Can a Dirichlet series have infinitely many zeros and be nonzero? To be precise, by a Dirichlet series I mean a function of the form $s\mapsto \sum_{n\geq 1}\frac{a_n}{n^s}$ where the domain is the ...
• 165
55 views

### How do we increase the region of convergence for the Riemann Zeta function (using Dirichlet Series form)?

The Riemann Zeta Function can be defined as: $\zeta(s)=\sum \frac 1 {n^s}$ for $s>1$. The series converges for $s>1$. wiki (https://en.wikipedia.org/wiki/Riemann_zeta_function) mentions that: An ...
1 vote
47 views

### Manipulating Dirichlet series generating functions

This is from p.$61$ in Wilf's "generatingfunctionology" As a step to solving for the $b$'s in terms of the $a$'s Given: $a_n = \sum_{d\mid n}b_d$ Consider the Dirichlet power series ...
• 137
44 views

1 vote
61 views

• 3,887
### What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?
The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by $$A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p$$ is this serie calculated ...