Ramanujan tau function
Determining values of the Ramanujan tau function directly can be somewhat involved. For example, the values of , , and will be determined:
To determine , , and , we need to find the coefficient of , , and , respectively, of the expression
Note that we only need to consider and , since higher values of yield powers (http://planetmath.org/Power) of that are too large. Thus:
Hence, , , and .
The sequence appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/A000594A000594.
Although the values of seem to increase rapidly as increases, the conjecture that for all has not yet been proven. This conjecture is known as Lehmer’s conjecture.
The Ramanujan tau function has the following properties:
It is a multiplicative function: For with , we have .
For any prime and any ,
For any prime ,
For any ,
For any and for any nonnegative integer which is a quadratic residue modulo ,
|Title||Ramanujan tau function|
|Date of creation||2013-03-22 17:51:24|
|Last modified on||2013-03-22 17:51:24|
|Last modified by||Wkbj79 (1863)|
|Synonym||Ramanujan’s tau function|