Timeline for multiplicative functions of powers
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 25, 2015 at 20:13 | comment | added | Igor Rivin | @so-calledfriendDon If you make your comment an answer, I will accept it. | |
| Oct 25, 2015 at 18:18 | comment | added | Igor Rivin | @so-calledfriendDon That was precisely the sort of thing I was asking about! I will check out the Wirsing reference. | |
| Oct 25, 2015 at 18:01 | comment | added | so-called friend Don | Perhaps I don't understand the question. But $\tau(n^k)$ is a nonnegative multiplicative function. And there are very general theorems giving asymptotics for partial sums of nonnegative multiplicative functions; e.g., there is a beautiful theorem of Wirsing that immediately applies to this question. See Wirsing, Eduard Das asymptotische Verhalten von Summen über multiplikative Funktionen. (German) Math. Ann. 143 1961 75–102. (But I think Wirsing is "overkill" here, in the sense that Brad Rodgers's suggestion above also works and is simpler.) | |
| Oct 25, 2015 at 10:06 | comment | added | Igor Rivin | @AlexeyUstinov Yes, I am aware of this reference. $f$ is an irreducible polynomial, and the result is a bound, not asymptotic. | |
| Oct 25, 2015 at 1:29 | comment | added | Alexey Ustinov | P. Erdos. "On the sum $\sum^{x} _{k=1}d( f (k))$. J. London Math. Soc. 27 (1952), 7–15. | |
| Oct 24, 2015 at 22:19 | comment | added | Igor Rivin | Thanks. The multiplicative function I had in mind when asking the question was $\tau$ (the number of divisors). There are many estimates for sums of $\tau(P(n)),$ but no asymptotics that I can find (with the possible exception of $P(n) = n^2,$ though even that is not clear), which might just mean that the question is hard. | |
| Oct 24, 2015 at 21:51 | review | First posts | |||
| Oct 24, 2015 at 21:55 | |||||
| Oct 24, 2015 at 21:49 | history | answered | asdasdasd | CC BY-SA 3.0 |