Questions tagged [theta-functions]
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145 questions
2 votes
1 answer
294 views
Show $\alpha(n)=0$ when $n \equiv 3\mod4$
Consider the modular forms $$\theta(q):=\sum_{n\in\mathbb Z}q^{n^2},\qquad E_4(q):=1+240\sum_{n\ge1}\sigma_3(n)q^n,$$ $$\sum_{n\ge1}\alpha(n)q^n:=\frac1{16}(\theta^{10}(q)-\theta^2(q)E_4(q^2)).\tag{$\...
- 341
3 votes
1 answer
206 views
Mellin transform of theta function directly gives Casimir energy for massive scalar field?
Consider the theta-style function with $\nu \in \Bbb Z$ $$f_{\nu}(x)=\vert \log x \vert^\nu \sum_{n\in \Bbb N} e^{\frac{n^2}{\log x}}$$ and the Mellin transform $$ F_{\nu}(r)=\int_{(0,1)} (1+2f_{\nu}(...
- 621
4 votes
0 answers
127 views
counting integral points on grassmannians by modular forms
An integral point $P$ of a Grassmannian $Gr(k,n)$ is a $k$-dimensional subspace such that $P \cap \mathbb{Z}^n$ is a rank $k$ sublattice of $\mathbb{Z}^n$. Its height $H$ is given by the determinant ...
- 439
4 votes
2 answers
276 views
Transformation law for Dedekind $\eta$ from that of Jacobi's $\vartheta$
The Dedekind $\eta$ function has the transformation law $$\eta(-1/\tau) = \sqrt{-i\tau}\,\eta(\tau)\ .\tag{1}$$ The Jacobi $\vartheta$ functions obey very similar laws, e.g. $$\vartheta_{01}(z; -1/\...
- 1,293
0 votes
0 answers
204 views
The starting point of Riemann connecting the plane curve and theta function
While reading and searching various references, I found out that a mathematician named Plucker discovered that there are 28 bitangents of quartic curves for plane curves, and that a mathematician ...
- 339
19 votes
2 answers
837 views
CM zeros of unary theta series
Show that $$\sum_{n\ge1}\left(\dfrac{n}{5}\right)\exp(2i\pi n^2(18+i)/50)=0$$ where $(n/5)=1,-1,-1,1,0,...$ is the Legendre symbol. The series converges extremely rapidly, so this can be checked ...
- 13.9k
1 vote
0 answers
195 views
strange duality and factorization of generalied theta functions
Recall the celebrated rank-level strange duality isomorphism: $$H^0(SU_C(r),L^l) \to H^0(U_C(l), \mathcal{O}(r\Theta))$$ between level $l$ generalized theta functions on the moduli space of rank $r$ ...
- 3,747
10 votes
1 answer
650 views
Special phenomenon of zeroes of Dirichlet theta function
Let $e(z)=e^{2\pi i z} $ ; $ \chi_q$ is a primitive Dirichlet character mod q , $\nu=\frac{1-\chi(-1)}{2}$ and $ \theta(z,\chi_q)$ is the Dirichlet theta function : $$\theta(z,\chi_q)=\frac12 \...
- 341
4 votes
0 answers
305 views
Could this closed-form expression for the integral of the Riemann $\xi$-function along the critical line provide new insights?
The Riemann $\xi$ and $\Xi$-functions are respectively defined as: \begin{align} \xi(s) &= \frac{s\,(s-1)}{2}\, \pi^{-s/2} \,\Gamma\left(\frac{s}{2}\,\right) \zeta(s) \qquad s \in \mathbb{C} \\ \...
- 169
3 votes
0 answers
110 views
Congruences regarding $4n$-dimensional lattices
A sequence of integers $(a_n)_{n\geq 1}$ satisfies Gauss congruence if $$\sum_{d\mid n}\mu(d)a_{n/d}\equiv 0\pmod{n}$$ for every $n\geq 1$. Such sequences are also called Dold sequences, Newton ...
- 211
3 votes
0 answers
161 views
Using Ramanujan's cubic continued fraction $C(q)$ and $x^3+y^3=1$ to solve the Bring quintic?
The octic Ramanujan-Selberg continued fraction $S(q)$ and $x^8+y^8=1$ can solve the Bring quintic. So can the Rogers-Ramanujan continued fraction $R(q)$ and $x^5+y^5=1.\,$ It turns out Ramanujan's ...
- 13k
2 votes
0 answers
168 views
Two new formulas to solve the Bring quintic using only Jacobi $\vartheta_3(q)$ and $\vartheta_4(q)$?
(Note: Emil Jann Fiedler found the formula for the Bring quintic using $R(q)$ in 2021, and these two formulas using $\vartheta_3(q)$ and $\vartheta_4(q)$ in 2022.) Recall the Jacobi theta functions, $$...
- 13k
3 votes
0 answers
321 views
Proving the Lambert series of $\theta^8$ and Eisenstein series $E_4$
This is a cross-post from MSE since there wasn't having enough attention. I need your help on proving the following identity Theorem Let $q=e^{-\pi \frac{K'}{K}}$ where $K$ denotes the complete ...
- 101
5 votes
0 answers
374 views
Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
- 13k
1 vote
0 answers
225 views
On the Jacobi theta functions and the Borweins' cubic theta functions
The post has been divided into sections to show some patterns, as well as possible evaluations of, $$_2F_1\big(s,1-s,1,z\big)$$ with $s = \frac12, \frac13, \frac14, \frac16$ for infinitely many ...
- 13k
9 votes
2 answers
2k views
Why does this theta function value yield such a good Riemann sum approximation?
Let $T(q)$ denote the third Jacobi theta function at $z=0$; i.e., $$T(q) := \sum_{n\in\mathbb{Z}} q^{n^2}.$$ Then for any $x>0$, $T(\exp(-1/x)) \approx \sqrt{\pi x}$. For example, as the entry for ...
- 88.1k
-6 votes
1 answer
204 views
Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]
Consider the series defined by \begin{equation} f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))} \end{equation} is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
5 votes
2 answers
431 views
Constant term arising in general exact expression for the canonical height of a Mordell-Weil generator point on an elliptic curve $E$
First I shall begin by laying out some notation (I shall be using the conventions that are used by both DLMF and Mathematica which occasionally differ from the standard literature): Let $\Lambda:=\...
- 541
4 votes
0 answers
140 views
Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me just pick one of them (complete elliptic integral of the first kind [1]): $$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
- 5,748
2 votes
0 answers
102 views
How to write the division values of $\operatorname{sn}(u;k)$ as rational functions of theta functions with zero argument?
Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows: $$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$ $$\...
- 317
4 votes
1 answer
339 views
Is the left derivative of this theta function zero at $1$?
Is it true that $$f(x):=\sum_{j=1}^\infty(-1)^j j^2 x^{j^2}\to0$$ as $x\uparrow1$? (One may note that $f(x)=xh'(x)$, where $h(x):=\vartheta _4(0,x)/2$ and $\vartheta _4$ is a theta function, so that $...
- 143k
5 votes
0 answers
168 views
Ratio of theta functions as roots of polynomials
I already asked the same question here, but received no answer. I did some little progress and so I'm asking again. I was playing with the theta functions with argument $ z = 0 $ $ \vartheta_2(q) =\...
- 557
2 votes
0 answers
138 views
Ratio of theta function derivatives with theta function
I have the following ratios I want to compute. $$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial v} \right)^2 }{C + \left(\vartheta_3(v, q)\right)^2 }, $$ where $C$ is a constant. $$ \frac{ \...
- 159
2 votes
1 answer
197 views
Reference for modularity of the Andrews–Gordon–Rogers–Ramanujan identities?
The right-hand side of the identity https://mathworld.wolfram.com/Andrews-GordonIdentity.html is a $q$-series $\frac{(q^i,q^{2k+1-i},q^{2k+1};q^{2k+1})_\infty}{(q;q)_\infty}$; is there a reference of ...
- 824
1 vote
0 answers
130 views
Reference for the asymptotic mixing time of the random walk on the cycle
In Diaconis's book Group Representations in Probability and Statistics, Chapter 3C, there are explicit computations for the mixing time of the random walk on the cycle graph $\mathbb{Z}_{p}$, with $p$ ...
- 111
3 votes
0 answers
95 views
Additional symmetries in Theta-like function
cross-posted from https://math.stackexchange.com/questions/4708694/curious-symmetry-in-a-theta-like-function Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows $$ \...
- 551
1 vote
1 answer
382 views
The theta function of an odd Dirichlet character
The theta function $\theta_\chi(t)$ of a Dirichlet character $\chi$ is defined to be $\theta_\chi(t) = \frac{1}{2} \sum\limits_{n=-\infty}^\infty \chi(n) e^{2\pi i n^2 t}$ if $\chi(-1) = 1$ (i.e., $\...
- 536
2 votes
1 answer
268 views
2D lattice sum with numerator
I've been struggling a bit with a double sum that arose as the trace of an operator: $$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$ where $n$ is an even natural number. Is there ...
- 31
2 votes
0 answers
235 views
Theta characteristics of surfaces and theta functions
A theta-characteristic of a Riemann surface is commonly defined as a spinor bundle, i.e., a holomorphic line bundle whose square is the canonical line bundle. These are in a natural one-to-one ...
- 9,865
5 votes
3 answers
557 views
Generalizing Klein's order 7 formula $y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j$ to order 13?
I. Level 7 In Klein's "On the Order-Seven Transformations of Elliptic Functions", he gave two elegant resolvents of degrees 8 and 7 in pages 306 and 313. Translated to more understandable ...
- 13k
3 votes
0 answers
136 views
Theta lifting over function fields
Let $F$ be a number field and $\mathbb{A}$ its adele ring. For a dual reductive pair $G$ and $H$, let $\pi$ be a cuspidal irreducible representation of $G(\mathbb{A})$. Let $\Theta(\pi)$ be the global ...
- 1,759
2 votes
0 answers
144 views
Eta product of squared tau function
The Ramanujan tau function is the coefficient of the 24th power of the Dedekind eta function. $$ \eta(x)^{24}= x\prod_{m=1}^\infty (1 - x^m)^{24} =\sum_{n=1}^\infty \tau(n)\,x^n , $$ I want to know ...
- 365
4 votes
1 answer
228 views
Evaluation of mock modular forms at elliptic points
The holomorphic function $$F(\tau)=-\frac{1}{\vartheta_4(\tau)}\sum_{n\in\mathbb Z}\frac{(-1)^nq^{\frac{n^2}{2}-\frac 18}}{1-q^{n-\frac12}}=2q^{\frac38}(1+3q^{\frac12}+7q+14q^{\frac32}+\dots),$$ is a ...
- 99
3 votes
1 answer
378 views
Siegel modular forms in Mathematica
Is there a convenient way to work with Siegel modular forms in Mathematica? I am interested in doing analytic computations using the $\chi_{10}(\Omega)$ Siegel modular form, where $\Omega$ is the $2\...
- 491
10 votes
3 answers
665 views
On the Klein quartic and the similar $a^2b+b^2c+c^2a$?
Given the Ramanujan theta function, $$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$ Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}$. I. Degree 5 \begin{align} a &= q^{11/...
- 13k
1 vote
0 answers
209 views
What pre knowledge does Mumford's Tata collections on theta need?
I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...
- 11
6 votes
1 answer
472 views
What is the relationship between the Leech lattice and Dedekind eta function?
Like this old question, A conceptual proof of Jacobi's product formula for $\Delta$ ?, I am asking again for a conceptual proof of Jacobi's miraculous product formula for $\Delta$ (the unique ...
- 1,235
3 votes
1 answer
214 views
Entire function with almost periodic boundary condition?
Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...
- 73
5 votes
0 answers
201 views
Higher Cardano formulae in terms of $\Theta$
Consider a polynomial in one variable with complex coefficient $$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...
- 5,748
3 votes
1 answer
341 views
Equation about Jacobi Theta Functions
Reading some Conformal Field Theory, I came across the following equation about the Jacobi Theta functions without any justification: Let $$\theta_{2}(q)=\sum_{n \in \mathbb{Z}}q^{(n+\frac{1}{2})^{2}}$...
- 31
3 votes
0 answers
425 views
How to prove Gauss's identities on the action of the theta operator $f' = x\frac{\mathrm df}{\mathrm dx}$ on Jacobi theta functions?
(I have previously posted this question on Math StackExchange, but after getting no response there, I decided to ask it again here. Here is a link to the same question there, so if this question is ...
- 2,497
4 votes
0 answers
401 views
Geometric interpretation of Theta functions and the Jacobi inversion problem
A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
4 votes
0 answers
204 views
How to obtain the harmonic theta series via the global theta correspondence explicitly?
I am interested in the following kind of modular forms: let $(L,q)$ be an even unimodular lattice inside $V:=L\otimes \mathbb{Q}$, and $P$ is a harmonic homogeneous polynomial of degree $d$ on $\...
- 411
6 votes
1 answer
422 views
How to work out this elliptic function?
Let $f(x) = \sum\limits_{(n,m)\in\mathbb{Z}^2} \frac{1}{(x+ n + i m )^2}$ If feel it should be $1/E(x)$ where $E$ is some elliptic function, like $sn^2$. But Wolfram Alpha is giving me some strange ...
- 255
3 votes
0 answers
111 views
Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety
Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$. Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual. As usual, let $K(L):=\ker(\...
- 3,346
15 votes
1 answer
664 views
Geometric explanation for coincidence in lengths of 16-dimensional even unimodular lattices?
Question Up to equivalence, there are two positive-definite even unimodular lattices in $16$ dimensions: $D_{8}^{+}\oplus D_{8}^{+}$ and $D_{16}^{+}$. As observed by Witt in 1941, the theory of ...
- 411
1 vote
1 answer
151 views
Estimating two dimensional theta function
My feeling is that this should be written somewhere but I don't know what to search for. Let $Q(x,y)$ be a binary quadratic form over $\mathbb{C}$, with $\operatorname{Re}(Q)$ positive definite. Then ...
- 2,198
10 votes
2 answers
1k views
Reference on the Chern-Simons theory and WZW models for mathematicians
I would like to ask if there are any beginner friendly references for learning CS theory and WZW models. It seems that most mathematical texts on the subjects begin with convenient definitions that ...
- 71
3 votes
3 answers
781 views
Infinite product of $1-q^{n^2}$
Is there anything known about the following product? Is it a known function or related to a known function? $$\prod_{n\geqslant1}(1-q^{n^2})$$
- 549
6 votes
1 answer
379 views
Approximation for a series involving the derivative of a Jacobi theta function
I’ve considered the diffusion equation $$\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>...