6
votes
1answer
159 views

The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the ...
1
vote
1answer
55 views

Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$ \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1} $$ I have an intuitive ...
0
votes
0answers
27 views

About a specific mathematical series which is a power of the exponential function

My professor wrote the below exponential function just out of the box when he suggested a kernal for a 1D domain. $f(x) = e^{-\Big(a_1x_1+ \dfrac{1}{2} a_2 x_2^2 + \dfrac{1}{3} a_3 x_3^3 + .... ...
8
votes
0answers
313 views

$f'/f\in\mathbb{Z}[[x]]$ for polynomials vs. formal power series $f$

I am curious about the following problem from MIT's Problem Solving Seminar (#26 here, though the link may stop working after a few weeks): Let $f(x) = a_0+a_1x+\cdots\in\mathbb{Z}[[x]]$ be a ...
0
votes
0answers
105 views

Zeros of a power series

Suppose we have a power series with (real or complex) coefficients $\sum_{n \geq 0} a_n x^n$ (that has nonzero radius of convergence). Can one say something about its zeros in terms of the ...
4
votes
0answers
146 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
6
votes
1answer
107 views

Recognizing if a power series is a $q$-expansion of a modular form

Given a power series in $q$, is it possible to tell if it is the $q$-expansion of a modular form (of level $N$ say)? I don't need to show results of this sort, but it has come up enough that I'm ...
4
votes
1answer
96 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
82 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 $S_r(n)$ ...
13
votes
2answers
313 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
252 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
201 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
249 views

Calculating powers of 2 on a 2D grid without factoring.

Consider the following 2D infinitely large grid where the dots represent infinity: ...
4
votes
1answer
240 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
134 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
205 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 $0$. ...