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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 striclty 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!

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    $\begingroup$ Probabilistic heuristics suggest that the set of such Fourier coefficients (or equivalently, the set of non-trivial integer solutions to $m_1^4 + n m_2^4 = m_3^4 + n m_4^4$) is very sparse (only about $O(\log X)$ such solutions up to height $X$). So it is unlikely that analytic methods will be of much help here. Maybe there is some algebraic number theory approach but it doesn't look too promising (e.g. I don't see a norm form or other obviously algebraic structure here). $\endgroup$
    – Terry Tao
    Commented Feb 24, 2015 at 21:12
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    $\begingroup$ For what it's worth, the smallest number expressible as a sum of two fourth powers in two (genuinely) different ways is $635318657=133^4+134^4=59^4+158^4$. Euler knew this equation, Leech proved it's the smallest example, according to D1 in Guy, Unsolved Problems In Number Theory. $\endgroup$
    – Gerry Myerson
    Commented Feb 24, 2015 at 22:36
  • $\begingroup$ 1. You consider convergent or formal series? $\endgroup$
    – Sergei
    Commented Feb 25, 2015 at 8:49
  • $\begingroup$ 2. May you give a reference please where such theta-like functions are studied. $\endgroup$
    – Sergei
    Commented Feb 25, 2015 at 8:51
  • $\begingroup$ @Sergei: i have not such a reference. Indeed(but i apologize if it was not clear) i posted to know if such series have been already studied and if they have some non trivial property. Anyway i wrote those series as formal series(motivated by the reason explaned in the comment of Terry above), but i wrote here because i had the hope to threat them with analytic tools, and so i was wondering what is known about such functions, as asked in the last questions. $\endgroup$
    – Marcel1994
    Commented Feb 25, 2015 at 11:19

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