All Questions
Tagged with dirichlet-series summation
13
questions
1
vote
1
answer
184
views
Finding the sum of a series using a Fourier series
I am stuck on how to calculate the value of the following sum:
$\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$
I am aware that you need to find the corresponding function whose Fourier series is represented ...
- 351
2
votes
1
answer
105
views
How to find the sum of this infinite series
I am not sure how to evaluate the infinite sum:
$$\sum_{n=0}^\infty \frac{1}{(2n+1)^6}$$
Apparently, I can shift it to
$$\sum_{n=1}^\infty \frac{1}{(2n-1)^6}$$
which is supposed to be a well known sum ...
- 21
3
votes
1
answer
88
views
Issue proving a dubious identity involving Dirichlet series
Let $a_{1}, a_{2}, \ldots\in\mathbb{C}$ be a sequence, and let
$$ F(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}} \qquad\text{ and }\qquad G(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{(n+1)^{s}}. $$
...
- 2,461
2
votes
1
answer
111
views
Product over the primes with relation to the Dirichlet series
What is the value of $\displaystyle \prod_p\left(1+\frac{p^s}{(p^s-1)^2}\right)$
I got this product by defining a function $a(n)$ such that $a(n)=a(p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_n^{a_n})=a_1a_2a_3......
- 73
1
vote
0
answers
77
views
Proof of Vaughan's identity, based on comparing the Dirichlet coefficients
I am trying to figure out how to prove $- \frac{\zeta'}{\zeta}(s) = F(s) -G(s)\zeta'(s) -F(s)G(s)\zeta(s) -(\zeta(s)G(s) -1) \Big{(} - \frac{\zeta'}{\zeta}(s) - F(s) \Big{)}$ to prove $\Lambda(n) = ...
- 1,178
1
vote
1
answer
77
views
Why does this equality Dirichlet series hold?
Following on from my question here, I have hit a second roadblock.
I am working (very slowly!) through a paper here that demonstrates Riemann's analytic continuation of the zeta function $\zeta(s)=\...
1
vote
1
answer
75
views
What is meant by $\sum_{d \le x}f(d)$ in this equation?
Wikipedia's page (here) on the average order of arithmetic functions gives the following as a means of finding such an order using Dirichlet Series:
Define $f$ as an arithmetic function on $n$, and ...
1
vote
0
answers
122
views
Questions related to globally convergent formulas for the Dirichlet eta function $\eta(s)$
This question is about the relationship between the known globally convergent formula illustrated in formula (1) below and the conjectured formula illustrated in formula (2) below which I originally ...
- 7,631
0
votes
1
answer
42
views
Alternating Bertrand series
It is known that
$$\frac{\partial^n\zeta(s)}{\partial s^n}=(-1)^n\sum_{k=1}^\infty{\frac{\log^nk}{k^s}}$$
Can the following alternating version of the sum be expressed in terms of well-known ...
- 3,557
3
votes
1
answer
76
views
On a lemma by Newman relating summability and convergence
On page 73 of his book on Analytic Number Theory, Newman presents the following lemma:
Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^\infty \frac{a_n}{n}$ exists and $a_n + \log n$ is ...
- 825
0
votes
1
answer
203
views
Solving for variable inside a sum
So a few calcuations have ultimately led me to this expression
$$ \sum_{n=1}^\infty \frac{B_n^2\left( \sinh\left( \sqrt 2\, \pi n \right) - \sqrt 2\,\pi n \right)}{4\pi\sqrt 2\, n} = 1 $$
Is there ...
- 43
1
vote
0
answers
18
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Prove or disprove the existence of f(k) when $\sum_{n=1}^m{\frac{a_n}{n^s}}=\sum_{n=1}^m\frac{a_n+f^n(x)}{n^s}$.
Can we find a function, $f(k)$, such that $$\frac{a_n}{n^s}+\frac{a_{n+1}}{(n+1)^s}+\frac{a_{n+2}}{(n+2)^s}=\frac{a_n+x}{n^s}+\frac{a_{n+1}+f(x)}{(n+1)^s}+\frac{a_{n+2}+f(f(x))}{(n+2)^s}$$ for some $x$...
- 3,557
13
votes
3
answers
350
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Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$
For some time I've been playing with this kind of sums, for example I was able to find that
$$
\frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right)
$$
where
$$
\beta(x)=\sum_{k=...
- 5,627