I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is greater than 1( The additive claim that i hope to reach is clear to the reader: but here i'm interested in seeing if looking at this claim from the point of view of fourier series can give some insights).
I've thought to use Poisson summation formula on $e^{2 \pi x^4 t}$ and $e^{2 \pi nx^4 t}$ and hoping that writing the series in an another way i can find some new information and get the result mentioned above.
But before start to do calculation and experiments i felt that, since these kind of series(is similar to a theta series) have been surely well studied, it may be wiser to start asking:
What is known about such functions? Wich kind of estimates could be sufficient to reach the above thesis?
Thanks in advance!
Your function $f(t)$ is a product of two infinite series so we can apply the formula that gives the series product see https://en.wikipedia.org/wiki/Cauchy_product. In other words, for fixed $n$ and $t,$ put $\alpha(m)=e^{2i\pi m^2 t}$ and $\beta(l)=e^{2i\pi l^2 n t}=\alpha(l)^n$ then $f(t)=\sum_{m=0}^{+\infty}\alpha(m) *\sum_{l=0}^{+\infty}\alpha(l)^n$ which is equal to $f(t)=\sum_{m=0}^{+\infty} \sum_{c=0}^{m}\alpha(c)*\alpha(m-c)^n.$ So, we have $f(t)=\sum_{m=0}^{+\infty} \sum_{c=0}^{m} e^{2i\pi(c^2+(m-c)^2 n)t}.$I hope that this formula can help you.