  The ternary Goldbach conjecture states that every odd number $n\geq 7$ is the
sum of three primes. The estimation of the Fourier series $\sum_{p\leq x}
e(\alpha p)$ and related sums has been central to the study of the problem
since Hardy and Littlewood (1923). Here we show how to estimate such Fourier
series for $\alpha$ in the so-called major arcs, i.e., for $\alpha$ close to a
rational of small denominator. This is part of the author's proof of the
ternary Goldbach conjecture. In contrast to most previous work on the subject,
we will rely on a finite verification of the Generalized Riemann Hypothesis up
to a bounded conductor and bounded height, rather than on zero-free regions. We
apply a rigorous verification due to D. Platt; the results we obtain are both
rigorous and unconditional. The main point of the paper will be the development
of estimates on parabolic cylinder functions that make it possible to use
smoothing functions based on the Gaussian. The generality of our explicit
formulas will allow us to work with a wide variety of such functions.
