The approximate functional formula for the theta function and Diophantine Gauss sums

Abstract:

We consider the polygonal lines in the complex plane $\Bbb {C}$ whose $N$-th vertex is defined by $S_N = \sum _{n=0}^{N ’} \exp (i\omega \pi n^2)$ (with $\omega \in \Bbb {R}$), where the prime means that the first and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small $\omega$, to Cornu spirals (C-spirals), we prove the precise renormalization formula \begin{equation} \begin {split} &\left | \sum _{k=0}^{N} ’ \exp (i\omega \pi k^2) -\frac {\exp (sgn(\omega )i\pi /4)}{\sqrt {|\omega |}} \sum _{k=0}^n ’ \exp (-i\frac {\pi }{\omega } k^2)\right |\\ &\qquad \leq C \left |\frac {\omega N - n}{\omega }\right |, 0<|\omega | <1, \end{split} \end{equation} where $N=[[n/\omega ]]$, the nearest integer to $n/\omega$ and $1<C<3.14$ . This formula, which sharpens Hardy and Littlewood’s approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map \begin{equation} \omega \rightarrow -\frac {1}{\omega } \pmod 2 , \omega \in ]-1,+1[ \setminus \{0\}, \end{equation} whose orbits are analyzed by expressing $\omega$ as an even continued fraction.