Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.

The elementary method of Balog and Ruzsa and the large sieve of Linnik are utilized to investigate the behaviour of the 𝐿 1 norm of an exponential sum over the primes. A new proof of a lower bound due to Vaughan for the 𝐿 1 norm of an exponential sum formed with the von Mangoldt function is furnished.

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