Timeline for Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
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| Feb 25, 2015 at 11:19 | comment | added | Marcel1994 | @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. | |
| Feb 25, 2015 at 8:51 | comment | added | Sergei | 2. May you give a reference please where such theta-like functions are studied. | |
| Feb 25, 2015 at 8:49 | comment | added | Sergei | 1. You consider convergent or formal series? | |
| Feb 24, 2015 at 22:36 | comment | added | Gerry Myerson | 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. | |
| Feb 24, 2015 at 21:12 | comment | added | Terry Tao | 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). | |
| Feb 24, 2015 at 19:54 | history | edited | Marcel1994 | CC BY-SA 3.0 |
added 9 characters in body
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| Feb 24, 2015 at 19:42 | review | First posts | |||
| Feb 24, 2015 at 19:47 | |||||
| Feb 24, 2015 at 19:41 | history | asked | Marcel1994 | CC BY-SA 3.0 |