Timeline for Sum of series $a^{i^2}$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 15, 2015 at 13:07 | comment | added | Geoff Robinson | I understand that point, as I understand the link between Jacobi's work and theta functions. | |
| Jan 15, 2015 at 12:33 | comment | added | Hjalmar Rosengren | I you insist. But to me that is like saying $\cos(z)$ is the fourth root of the known function $\cos^4(z)$. Theta functions are quite fundamental and appear in lots of contexts apart from sums of four squares. | |
| Jan 15, 2015 at 12:14 | comment | added | Geoff Robinson | @HjalmarRosengren : OK, thanks, so $1+2f(a)$ is the 4-th root of a "known" function. | |
| Jan 15, 2015 at 7:24 | comment | added | Hjalmar Rosengren | Actually, Jacobi's $r_4(n)$ refers to squares of integers, not positive integers, so it's $(1+2f(a))^4=\sum_{n=0}^\infty r_4(n)a^n$. That's an essentially different problem. As Sergei Points out, $1+2f(a)$ is a theta function, and Jacobi's work on 2, 4, 6 and 8 squares is based on exploiting this fact. | |
| May 21, 2014 at 11:20 | vote | accept | user51031 | ||
| May 21, 2014 at 9:02 | comment | added | Geoff Robinson | @BrendanMcKay : I know what you mean, but the spirit of the question seems to be to describe the given series in terms of something "more familiar" | |
| May 21, 2014 at 6:44 | comment | added | Brendan McKay | Is $r_4(n)$ more "known" than $a^{n^2}$? | |
| May 20, 2014 at 17:02 | history | answered | Geoff Robinson | CC BY-SA 3.0 |