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 de Ramanujan" Sém. Delange–Pisot–Poitou (Théorie des nombres) , 9 : 14 (1967/68) pp. 1–17
[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)