All Questions
Tagged with dirichlet-series number-theory
184
questions
4
votes
1
answer
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
2
votes
1
answer
74
views
Dirichlet series and Laplace transform
Let $\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}$ be a Dirichlet series. It can be represented as a Riemann-Stieltjes integral as follows:
$$\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}=\int_1^\...
2
votes
1
answer
45
views
Perron's formula in the region of conditional convergence
I am a bit confused about the proof of Perron's formula. It states that for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ and real numbers $c > 0$, $c > \sigma_c$, $x > 0$ we have
$$\...
- 825
0
votes
1
answer
107
views
Proof of $\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx$ and $\limsup_{x\to\infty} \frac{\log|A(x)|}{\log x} = \sigma_c$
Theorem. Let $A(x) := \sum_{n\le x} a_n$. If $\sigma_c < 0$, then $A(x)$ is a bounded function, and $$\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx \tag{1}$$ for $\sigma > 0$. ...
- 14.2k
-1
votes
1
answer
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 ...
user1236730
1
vote
2
answers
104
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
2
votes
1
answer
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
4
votes
0
answers
74
views
Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
- 1,810
2
votes
1
answer
113
views
Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
0
votes
0
answers
40
views
LCM sum with $\log $'s
If I want to evaluate $$\sum _{[r,r']\leq x}\log r\log r'$$ I could write it as an integral using Perron's formula, pick up a pole, and get a main term which involves looking at (the derivatives at $\...
- 1,662
3
votes
1
answer
238
views
Alternating Dirichlet series involving the Möbius function.
It is well known that:
$$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$
with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function.
Numerical ...
- 3,191
1
vote
0
answers
40
views
Question on conjectured method of extending convergence of Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$
The question here is motivated by this Math StackExchange question and this Math Overflow question which indicate the evaluation of the Dirchleta eta function
$$\eta(s)=\underset{K\to\infty}{\text{lim}...
- 7,631
1
vote
1
answer
85
views
What’s the best bound on the Dirichlet coefficients of $\zeta(s-1)^2/\zeta(s)$
We have $\frac{\zeta(s-1)^2}{\zeta(s)} = \sum\limits_{n\ge 1} \frac{a_n}{n^s}$, where $a_n = \sum\limits_{d|n} \mu(d) \sigma_0(\frac{n}{d}) \frac{n}{d} = \sum\limits_{d|n} \phi(d) \frac{n}{d}$. Here $\...
- 3,887
4
votes
1
answer
161
views
What is the value of $L'(1,\chi)$ where $\chi$ is the non-principal Dirichlet character modulo 4?
I was trying to compute the following sum:
$$\sum_{n\le x}{\frac{r_2(n)}{n}}$$
where $r_2(n)=\vert\{(a,b)\in\mathbb{Z}^2:a^2+b^2=n\}\vert$. Using Abel's summation formula with $a_n=r_2(n)$, $\varphi(t)...
- 298
0
votes
1
answer
49
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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 ...
