math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 16: How many positive directions does the Archimedean place have? A counting law with a boundary phase

Contributed by Pedro Caceres

Restrict the archimedean part of Weil's quadratic form, for $\zeta$ or for a primitive real Dirichlet $L$-function of conductor $q$ and parity $\kappa$, to test functions supported in an interval of length $a$, and let $n_+(a)$ be the number of its positive directions, the directions that the pole and the primes must absorb. A Szeg\H{o}-type argument gives $n_+(a)=t_0a/\pi+o(a)$, where $t_0$ is the zero of $\vartheta'$. We observe that the form is $\Lambda-\psi_\kappa(-\Delta)$ with $\psi_\kappa$ a complete Bernstein function (proved) and, using the boundary phase of Kaleta, Kwa\'snicki and Ma{\l}ecki, we state, as a conjecture, the asymptotic law \[ n_+(a)=\Big\lfloor\frac{t_0a}\pi+\frac{2\varpi_\kappa(t_0)}\pi\Big\rfloor , \] with an explicit phase $\varpi_\kappa$ that depends only on the parity, the conductor entering only through $t_0$. We test it in seven cases, $\zeta$ and $\chi_{-3},\chi_{-4},\chi_5,\chi_{-7},\chi_8,\chi_{-8}$. On the finest mesh the shifts of the jumps $m=2,\dots,12$ bracket the prediction $2\varpi_\kappa(t_0)/\pi$, which ranges from $0.18$ to $0.59$, to within $1$ to $3\cdot10^{-3}$ for the characters and $3\cdot10^{-2}$ for $\zeta$. A second, independent implementation extrapolates in the mesh on three to nine jumps per case; the extrapolated shifts lie above the prediction by $0.2$ to $2.3\cdot10^{-4}$ in every case, a one-sided offset of the size of the extrapolation error, which we have checked to be a bias in one case only. The first jump is an exception, and for $\zeta$ the shift reaches its limit only after about ten jumps. A smoothed count removes the integer part and leaves ``degree plus constant'' with the boundary phase as constant; on a circle the constant is carried by the end points and cancels between an arc and its complement; and treating $-\log q$ as a degree, $q\leftrightarrow1/q$ acts on the band edge as $t_0\mapsto4\pi^2/t_0$ to leading order, with $\zeta$ at the self-dual point $t_0\approx2\pi$. We record what the law implies for the signature of Weil's form and where it stands with respect to the Riemann--Roch formulas for $\overline{\operatorname{Spec}\mathbb Z}$. The law is a conjecture supported by these computations. The Bernstein structure is proved here; the first-order count is quoted from Part~14 and its proof is not repeated.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 9 pages, 1 figure; Part 16 of the 47-part book-series Cross-Energy of Dirichlet Partial Sums (Parts 0-46), block III (The margin and the constant B-hat); formerly a standalone note (v1.0). Part 0 is the guide.