After playing with all sort of resonant circuits and increasing up to 6 my number of LEDs is series (2 green+ 2 red + 2 yellow), I observe that in any case, the cause of the effect is a capacitive coupling between each coil end and either the generator or the ground. The longer the coils, the wider their conductor surface and their ability to capacitive coupling. In any case, despite the appearance of an open circuit, the circuit is closed.
Current is drawn from the generator. When a resistance is placed in series with the generator output, the voltage drops.
It is easy to measure the power drawn from the generator by connecting 2 probes to this resistance terminals, measuring the voltage drop by using the oscilloscope in differential mode. From the voltage, we get the current with I=dU/R (dU being the voltage at the resistance) and the power by P=U*I. The power in the LEDs is always less than that provided by the generator.
I analyse such a device as an impedance adapter. The very high impedance at one end of the coil allows a very weak displacement current in air by capacitive effects, but due the high voltage and frequency (I can get up to 400V from a 15v generator) this current is enough to pass a power able to light several LEDs.
The higher the Q of the circuit, the higher the effect. So I had the idea to use a quartz crystal instead of a LC circuit. A quartz crystal has a much higher Q than an ordinary LC circuit.
An aluminum plate plays the role of the ground (it can be also directly connected to the ground, with just a little shift of the resonant frequency. On the photos, it is not). A quartz crystal of 2150 Khz is connected to this plate, and on the other side, to the classical circuit with 2x 1N4148 and the LED (see photo QuartzLed1.jpg). As there is no coil and the quartz is small, we lost the wide conductor surface acting as a capacity and have only a narrow separation between the two ends of the circuit, so I had to add a terminal capacity which is an iron sphere painted in blak (see QuartzLed2.jpg).
The signal from a transmitter is provided by a short "antenna" (which is not really an antenna), near the sphere capacity. It is to be emphasized that radiated HF is negligible due to the inefficiency of the antenna wich is far from being tuned to the wave length (around 20 cm, when a quater wave length of a tuned antenna would require 37 mtrs). So we work in the quasistatic approximation, meaning there is no propagation phenomena in air or along the conductors whose dimensions are too small in comparison with the wavelength, no plane wave, only an electric field.
We see that this setup lights also a LED. The signal must be very stable. A shift of +/-500Hz switch off the LED. In brief, I didn't see any anomalous results.