Questions tagged [theta-functions]
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145 questions
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Show $\alpha(n)=0$ when $n \equiv 3\mod4$
Consider the modular forms
$$\theta(q):=\sum_{n\in\mathbb Z}q^{n^2},\qquad E_4(q):=1+240\sum_{n\ge1}\sigma_3(n)q^n,$$
$$\sum_{n\ge1}\alpha(n)q^n:=\frac1{16}(\theta^{10}(q)-\theta^2(q)E_4(q^2)).\tag{$\...
- 341
3
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1
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230
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Mellin transform of theta function directly gives Casimir energy for massive scalar field?
Consider the theta-style function with $\nu \in \Bbb Z$
$$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}(...
- 137
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134
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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
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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.
$$\vartheta_{01}(z; -1/\...
- 1,293
0
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214
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The starting point of Riemann connecting the plane curve and theta function
While reading and searching various references, I found out that a mathematician named Plucker discovered that there are 28 bitangents of quartic curves for plane curves, and that a mathematician ...
- 339
19
votes
2
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880
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CM zeros of unary theta series
Show that
$$\sum_{n\ge1}\left(\dfrac{n}{5}\right)\exp(2i\pi n^2(18+i)/50)=0$$
where $(n/5)=1,-1,-1,1,0,...$ is the Legendre symbol. The series converges extremely
rapidly, so this can be checked ...
- 14.7k
1
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207
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strange duality and factorization of generalied theta functions
Recall the celebrated rank-level strange duality isomorphism:
$$H^0(SU_C(r),L^l) \to H^0(U_C(l), \mathcal{O}(r\Theta))$$
between level $l$ generalized theta functions on the moduli space of rank $r$ ...
- 3,757
10
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1
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Special phenomenon of zeroes of Dirichlet theta function
Let $e(z)=e^{2\pi i z} $ ; $ \chi_q$ is a primitive Dirichlet character mod q ,
$\nu=\frac{1-\chi(-1)}{2}$ and $
\theta(z,\chi_q)$ is the Dirichlet theta function : $$\theta(z,\chi_q)=\frac12 \...
- 341
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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} \\
\...
- 179
3
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118
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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
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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 ...
- 13.3k
2
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182
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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,
$$...
- 13.3k
3
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340
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Proving the Lambert series of $\theta^8$ and Eisenstein series $E_4$
This is a cross-post from MSE since there wasn't having enough attention.
I need your help on proving the following identity
Theorem
Let $q=e^{-\pi \frac{K'}{K}}$ where $K$ denotes the complete ...
- 101
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Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine
Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
- 13.3k
1
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On the Jacobi theta functions and the Borweins' cubic theta functions
The post has been divided into sections to show some patterns, as well as possible evaluations of,
$$_2F_1\big(s,1-s,1,z\big)$$
with $s = \frac12, \frac13, \frac14, \frac16$ for infinitely many ...
- 13.3k