Ramanujan function
The function , where
is the coefficient of
(
) in the expansion of the product
![]() |
as a power series:
![]() |
If one puts
![]() |
then the Ramanujan function is the -th Fourier coefficient of the cusp form
, which was first investigated by S. Ramanujan [1]. Certain values of the Ramanujan function:
,
,
,
,
,
,
,
. Ramanujan conjectured (and L.J. Mordell proved) that the following properties of the Ramanujan function are true:
![]() |
![]() |
Consequently, the calculation of reduces to calculating
when
is prime. It is known that
(see Ramanujan hypothesis). It is known that the Ramanujan function satisfies many congruence relations. For example, Ramanujan knew the congruence
![]() |
Examples of congruence relations discovered later are:
![]() |
![]() |
etc.
References
[1] | S. Ramanujan, "On certain arithmetical functions" Trans. Cambridge Philos. Soc. , 22 (1916) pp. 159–184 |
[2] | J.-P. Serre, "Une interpretation des congruences relatives à la function ![]() |
[3] | O.M. Fomenko, "Applications of the theory of modular forms to number theory" J. Soviet Math. , 14 : 4 (1980) pp. 1307–1362 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 15 (1977) pp. 5–91 |
Comments
It is still (1990) not known whether there exists an such that
. One believes that the answer is "no" . For an elementary introduction to the background of
, see [a1].
References
[a1] | T.M. Apostol, "Modular functions and Dirichlet series in number theory" , Springer (1976) |
Ramanujan function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramanujan_function&oldid=15321