On guessing whether a sequence has a certain property.
We study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form [g(n)].
Let , and be three distinct primitive holomorphic cusp forms of even integral weights , and for the full modular group , respectively, and let , and denote the th normalized Fourier coefficients of , and , respectively. We consider the cancellations of sums related to arithmetic functions , twisted by and establish the following results: for any , where , are any fixed positive integers.
H. P. F. Swinnerton-Dyer determined the structure of the ring of modular forms modulo p in the elliptic modular case. In this paper, the structure of the ring of Hilbert modular forms modulo p is studied. In the case where the discriminant of corresponding quadratic field is 8 (or 5), the explicit structure is determined.
In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit...