Questions tagged [theta-series]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
5 votes
1 answer
180 views

Approximation for a series involving the derivative of a Jacobi theta function

I’ve considered the diffusion equation $$\frac{\partial f(x,t)}{\partial t}=\frac12 \frac{\partial^2 f(x,t)}{\partial x^2}$$ with the conditions $f(x,0)=\delta(x)$ and $f(-1,t)=f(1,t)=0\ \forall t>...
8 votes
0 answers
246 views

Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$

Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$. For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...
4 votes
1 answer
228 views

Closed formula for reversion of Jacobi theta series

Considering the Jacobi theta: $\theta_3(z) = \sum_{n\in\mathbb{Z}} q^{n^2}$, we can invert $\theta_3-1$ in a small enough neighbourhood of 0. Routine computation with Lagrange-Burmann inversion gives ...
  • 303
4 votes
1 answer
289 views

What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?

Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
  • 601
4 votes
0 answers
58 views

$3$-variable Jacobi style identity linked to generalised Frobenius partitions

I was fiddling around with a family of probabilistic models and came across two "identities", which appear to be linked to generalized Frobenius partitions (more on this below). I would be ...
  • 552
5 votes
1 answer
189 views

Jacobi forms and Kato's modular units

this is pretty much just a silly literature question; apologies in advance. Kato uses the following theta function (or slight variants thereof) in his construction of his Euler system: $$\Theta(\tau, ...
  • 1,550
24 votes
2 answers
904 views

Which even lattices have a theta series with this property?

This is a slight generalization of a question I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this ...
  • 964
1 vote
0 answers
147 views

"Modularity" of generalized theta series

The most basic theta series is defined by $\theta(z)=\sum_{n=-\infty}^{\infty}q^{n^2}$, where $q=e^{2\pi iz}$. This is connected to the question of how many representatations does an integer $n$ have ...
  • 3,512
0 votes
0 answers
90 views

Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients

Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above: $$(i)\ 1 + 80q^3 ...
  • 3,104
2 votes
0 answers
99 views

Understanding proof of q-theta function expression

In arXiv:math/0309252v4 at the bottom of page 11, the following result is proposed $$ b_1 \theta(b_0 b_1^{\pm};p) \frac{z^{-1}\theta(z^2;p)}{\theta(b_0 z^{\pm};p)\theta(b_1 z^{\pm};p)} = (p;p)^{-2}\...
  • 181
7 votes
1 answer
157 views

(Binary) Theta functions and Atkin's U-operator

Let $D<0$ be a fundamental discriminant and consider the theta series $$\vartheta_Q(\tau)=\sum_{v\in\mathbb{Z}^2} q^{Q(v)}$$ associated to a quadratic form $Q$ of discriminant $D$. It appears to be ...
  • 189
2 votes
0 answers
89 views

Generalised theta series for fixed-rank sublattices

The theta series for a lattice $\Lambda$ is defined by $$\displaystyle \Theta_\Lambda(q) = \sum_{x \in \Lambda} q^{x \cdot x}.$$ Setting $q=e^{-\pi\tau}$ yields the (maybe more usual) related theta ...
  • 303
3 votes
1 answer
170 views

"Sparse" Theta Series

The number of integer points with a given norm in the integer grid $\mathbb{Z} \times \mathbb{Z}$ may be calculated via the generating function $$\theta_3(q)^2= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\...
  • 790
4 votes
0 answers
217 views

Expansion of Jacobi theta function at $p$-torsion

I am aware of the formula $$\Theta(z,q)=z\exp\left( -2\sum_{k\geq 1} \frac{z^{2k}}{(2k)!}E_{2k}(q)\right)$$ for the Jacobi theta function at the origin $z=0$. The definition I am using for the theta ...
7 votes
1 answer
333 views

Modularity of certain theta series associated to hyperbolic lattice

Let $L$ be an even hyperbolic lattice, i.e. a free $\mathbb{Z}$-module with a non-degenerate inner product $\cdot$ valued in $\mathbb{Z}$ of signature $(1,n)$ such that the norm of every vector is ...
  • 1,393
4 votes
0 answers
315 views

Are all weight 5/2 modular forms theta series?

Given a quadratic form in $5$ variables, the theta series is a weight $5/2$ modular form. For a weight $3/2$ modular form we know that there always is a linear combination of theta series that work ...
  • 2,341
9 votes
2 answers
1k views

Numerical coincidence? Why is $\sum(x^{k^2}) = \sum(x^{(k+1/2)^2})$ for $x = 0.8$?

A well-known formula for the logarithm is given by $$\log x = -\frac{\pi}{\operatorname{AGM}(a^2,b^2)}, \qquad x < 1$$ where AGM is the arithmetic-geometric mean, and $a$ and $b$ are given by $$a ...
  • 93
5 votes
0 answers
93 views

On a particular case of Dirichlet series [closed]

I've this series: $$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$ where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $. I need the limit of series like an analytic function of $...
  • 51
3 votes
1 answer
109 views

Number of vectors of fixed norm

Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols: $$ r_k(P)=\textrm{cardinality of }\{v\in \...
  • 2,214
3 votes
0 answers
81 views

Log-concavity of difference of theta functions

My knowledge on theta functions is limited, but I suspect that this is a quite challenging question. The 3rd Jacobian Theta function is given by \begin{equation} \theta_3(z,q)\,=\,\sum\limits_{n=-\...
  • 31
3 votes
0 answers
191 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
  • 31
6 votes
4 answers
473 views

Integral quaternary forms and theta functions

The following question arises when I attempt to understand the modular parameterization of the elliptic curve $$E:y^2-y=x^3-x$$ In Mazur-Swinnerton-Dyer and Zagier's construction, a theta function ...
  • 3,237
9 votes
2 answers
572 views

Determining the Lambert series for $xq+x^2q^4+x^3q^9+...+x^nq^{n^2}+...$

I am trying to determine the polynomials $P_n(x)$ from $$ xq+x^2q^4+x^3q^9+...+\ x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n}; $$ that is, $$ \sum_{d|n}x^{\frac nd-1}P_d(x)=\begin{cases}x^...
3 votes
1 answer
422 views

Modular property of indefinite degenerate theta series

Is there anything known about the (mock)modular properties, if any, of the following theta series, $\sum_{n\in {\mathbb Z}^r_+} e^{2\pi i \langle b, n\rangle} q^{\frac12 \langle n,n\rangle}$, where $...
3 votes
0 answers
77 views

Noncommutivity of various lifts

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$ for simplicity. Then an automorphic form on $D^{x}/F^{x}$ has several different interpretations. First, through ...
  • 2,341
4 votes
0 answers
96 views

Global theta series on GL

Let $E/F$ be a quadratic extension of number field and let $V$ be a Hermitian space over $E$. Then we have Weil representations for the dual pair $U(n,n)\times U(V)$, and we can consider the theta ...
  • 181
6 votes
1 answer
204 views

Recursions for some binary theta series in characteristic 3

Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of $k$...
  • 4,980