Timeline for Historical question in analytic number theory
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 29, 2010 at 18:46 | comment | added | maks | Btw, in Carlson's theorem you just need c < pi (and that is optimal, because of the function sin(pi*z)), not c < pi/2, as I wrote earlier. | |
| Jan 29, 2010 at 18:43 | comment | added | maks | Knowing the functional equation, at all the positive integers is enough to recover the functional equation for all complex z :) But the proof uses machinery well beyond Euler times, and a bound on the growth of the Riemann zeta function usually derived from the functional equation. In any case, by a theorem of Carlson, two entire functions, that don't grow faster than exp(c*|z|) (c < pi/2) (which is the case for (z-1)zeta(z) and hence a fortiori the functional equation term (z-1)2^z*pi^(z-1)*(blah blah)) and that agree on all positive integers, must be equal for all complex z. | |
| Jan 29, 2010 at 15:17 | comment | added | Emerton | Euler knew how to evaluate zeta at negative integers via abel summation, and of course also how to compute it at positive even integers. He observed some form of the functional equation relating the values, but didn't have an overall notion of zeta as a function of a complex variable, as far as I know. It seems possible that Riemann was also influenced by these ideas of Euler; does anyone know whether this is the case? | |
| Jan 29, 2010 at 7:23 | history | answered | maks | CC BY-SA 2.5 |