Harmonic Theta-Mode Dominance and Superlinear Real-Zero Growth for the Riemann–Jacobi Kernel

Abstract

This preprint studies the real-zero growth of high derivatives of the standard Riemann–Jacobi kernel associated with the Riemann Xi function. The main result proves that the number of distinct real zeros of the derivatives grows faster than any fixed linear multiple of the derivative order. The proof uses an exact reduction of each theta mode to a generalized Bell-polynomial family, followed by a saddle-point analysis that identifies the dominant oscillatory behavior of individual modes. A key feature is that different theta modes dominate on disjoint spatial regions. On each such region, the oscillations of the dominant mode can be transferred rigorously to the full infinite theta sum. The contribution from successive modes decreases only harmonically, so accumulating arbitrarily many fixed modes forces superlinear growth of the total number of real zeros. The argument also gives a precise connection with classical asymptotic theory for Bell and Touchard polynomials and with earlier work on oscillations generated by repeated differentiation. A sharper logarithmically enhanced asymptotic law is proposed as a conjecture, but is not used in the proof of the main theorem. The results concern the derivative geometry of the Riemann–Jacobi kernel itself. They do not assume, prove, or disprove the Riemann hypothesis.