The *theta function* is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum\_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the following important transformation property.



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> **Theta reciprocity**: $\theta(\tau)=\frac{1}{\sqrt{\tau}}\theta\left(\frac{1}{\tau}\right)$.
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This theorem, while fundamentally analytic—the proof is just Poisson summation coupled with the fact that a Gaussian is its own Fourier transform—has serious arithmetic significance.


* It is the key ingredient in the proof of the functional equation of the Riemann zeta function.
* It expresses the *automorphy* of the theta function.


Theta reciprocity also provides an analytic proof (actually, the *only* proof, as far as I know) of the Landsberg-Schaar relation


$$\frac{1}{\sqrt{p}}\sum\_{n=0}^{p-1}\exp\left(\frac{2\pi i n^2 q}{p}\right)=\frac{e^{\pi i/4}}{\sqrt{2q}}\sum\_{n=0}^{2q-1}\exp\left(-\frac{\pi i n^2 p}{2q}\right)$$


where $p$ and $q$ are arbitrary positive integers. To prove it, apply theta reciprocity to $\tau=2iq/p+\epsilon$, $\epsilon>0$, and then let $\epsilon\to 0$.


This reduces to the formula for the quadratic Gauss sum when $q=1$:


$$\sum\_{n=0}^{p-1} e^{2 \pi i n^2 / p} =
\begin{cases}
\sqrt{p} & \textrm{if } \; p\equiv 1\mod 4 \\\
i\sqrt{p} & \textrm{if } \; p\equiv 3\mod 4
\end{cases}$$


(where $p$ is an odd prime). From this, it's not hard to deduce Gauss's "golden theorem".



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> **Quadratic reciprocity**: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)(q-1)/4}$ for odd primes $p$ and $q$.
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For reference, this is worked out in detail in the paper "[Applications of heat kernels on abelian groups: $\zeta(2n)$, quadratic reciprocity, Bessel integrals](http://www.math.kth.se/~akarl/langmemorial.pdf)" by Anders Karlsson.




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I feel like there is some deep mathematics going on behind the scenes here, but I don't know what.



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> Why should we expect theta reciprocity to be related to quadratic reciprocity? Is there a high-concept explanation of this phenomenon? If there is, can it be generalized to other reciprocity laws (like Artin reciprocity)?
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Hopefully some wise number theorist can shed some light on this!



What do theta functions have to do with quadratic reciprocity?
Going in the direction of more generality:


With $\theta(\tau)=\sum\_n\exp(\pi i n^2 \tau)$, theta reciprocity describes how the function behaves under the linear fractional transformation $[\begin{smallmatrix} 0&1 \\ -1&0\end{smallmatrix}]$. From this one can show it's an automorphic form (of half integral weight, on a congruence subgroup). Automorphic forms and more generally automorphic representations are linked by the Langlands program to a very general approach to a non-abelian class field theory. Your "Why should we expect ..." question is dead-on. This is very deep and surprising stuff.


In the direction of more specificity, the connection to the heat kernel is fascinating. (In this context, Serge Lang was a great promoter of 'the ubiquitous heat kernel.') The theta function proof is also discussed in Dym and McKean's 1972 book *"Fourier Series and Integrals"* and in Richard Bellman's 1961 book *"A Brief Introduction to Theta Functions."* Bellman points out that theta reciprocity is a remarkable consequence of the fact that when the theta function is extended to two variables, both sides of the reciprocity law are solutions to the heat equation. One is, for $t\to 0$ what physicists call a 'similarity solution' while the other is, for $t\to \infty$ the separation of variables solution. By the uniqueness theorem for solutions to PDEs, the two sides must be equal!


A special case of quadratic reciprocity is that an odd prime $p$ is a sum of two squares if and only if $p\equiv 1\bmod 4$. This can be be done via the theta function and is in fact given in Jacobi's original 1829 book *"Fundamenta nova theoriae functionum ellipticarum."*


