Tagged Questions
3
votes
1answer
53 views
A numeral system built around Dirichlet series, by analogy of how positional numeral systems are built around power series?
For any natural number and chosen base p, the number admits a unique expression of the form $a_np^n + ... + a_2p^2 + a_1p^1 + a_0$, where $a_k < p$ for all k. This property is effectively what ...
1
vote
1answer
52 views
Coefficients of powers of the theta function
Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$
Now, I shall show that the powers of $\theta$ are given by
$$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$
where $S_r(n)$ ...
10
votes
2answers
199 views
How to do a very long division: continued fraction for tan
I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
6
votes
1answer
184 views
Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$
The following is a historical question, but first some background:
Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
2
votes
1answer
144 views
Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?
Let $x$ be a positive real number.
Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$.
Call this function $f(n,x)$.
Can we give good upper and lower bounds of $f(n,x)$ ...
0
votes
1answer
199 views
Calculating powers of 2 on a 2D grid without factoring.
Consider the following 2D infinitely large grid where the dots represent infinity:
...
3
votes
1answer
170 views
closed form for a series over the Riemann zeta zeros
given the series $ \sum_{\rho} \frac{1}{z-\rho} $ here the sum is taken OVER the roots of the Riemann function on the critical line $ 0 < Re(s) <1 $
the summation is understood as we sum the pair ...
2
votes
1answer
122 views
Reason behind the reciprocity of series
This question may appear to be a silly one for experts.
From long back I have been observing all kinds of series but every-series contain a reciprocal part, I mean the " one over something " , is ...
2
votes
3answers
195 views
What is the expression for this summation?
Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...
