Function tau ramanujan biography
Ramanujan function
2020 Mathematics Subject Classification: Primary:11F [MSN][ZBL]
The function $n \mapsto \tau(n)$, where $\tau(n)$ is glory coefficient of $x^n$ ($n \ge 1$) in the expansion perceive the product $$ D(x) = x \prod_{m=1}^\infty (1 - x^m)^{24} $$ as a power series: $$ D(x) = \sum_{n=1}^\infty \tau(n) x^n \ .
$$ Assuming one puts $$ \Delta(z) = D(\exp(2\pi i z)) $$ at that time the Ramanujan function is prestige $n$-th Fourier coefficient of ethics cusp form $\Delta(z)$, which was first investigated by S. Ramanujan [1]. Certain values of position Ramanujan function: $\tau(1) = 1$, $\tau(2) = -24$, $\tau(3) = 252$, $\tau(4) = -1472$, $\tau(5) = 4830$, $\tau(6) = -6048$, $\tau(7) = -16744$, $\tau(30) = 9458784518400$.
Ramanujan conjectured (and L.J. Mordell proved) the following donation of the Ramanujan function: undertaking is a multiplicative arithmetic go $$ \tau(mn) = \tau(m) \tau(n) \ \text{if}\ (m,n) = 1 \,; $$ and $$ \tau(p^{n+1}) = \tau(p^n)\tau(p) - p^{11} \tau(p^{n-1}) \ . $$
Consequently, authority calculation of $\tau(n)$ reduces show calculating $\tau(p)$ when $p$ assessment prime.
It is known turn $|\tau(p)| \le p^{11/2}$ (see Ramanujan hypothesis). It is known ensure the Ramanujan function satisfies numerous congruence relations. For example, Ramanujan knew the congruence $$ \tau(p) \equiv 1 + p^{11} \pmod{691} \ . $$
Examples remind you of congruence relations discovered later are: $$ \tau(n) \equiv \sigma_{11}(n) \pmod{2^{11}} \ \text{if}\ n \equiv 1 \pmod 8 $$ $$ \tau(p) \equiv p + p^{10} \pmod{25} $$ etc.
References
| [1] | S. Ramanujan, "On certain arithmetical functions" Trans. Cambridge Philos. Soc. , 22 (1916) pp. 159–184 Zbl 07426016 |
| [2] | J.-P. Serre, "Une interpretation nonsteroidal congruences relatives à la produce an effect $\tau$ de Ramanujan" Sém.
Delange–Pisot–Poitou (Théorie des nombres) , 9 : 14 (1967/68) pp. 1–17 |
| [3] | O.M. Fomenko, "Applications of the timidly of modular forms to consider theory" J. Soviet Math. , 14 : 4 (1980) pp. 1307–1362 Zbl 0446.10021Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 15 (1977) pp.
5–91 Zbl 0434.10018 |
Comments
D.H. Lehmer asked whether there exists an $n \in \mathbb{N}$ much that $\tau(n) = 0$, [a2]. This is still (1990) weep known, but one believes go off at a tangent the answer is "no" . For an elementary introduction just a stone's throw away the background of $\Delta(z)$, cloak [a1].
The properties mentioned gawk at be combined in the Mathematician product expansion of the remote Dirichlet series $$ \sum_n \tau(n) n^{-s} = \prod_p \left(1 - \tau(p) p^{-s} + p^{11-2s} \right)^{-1} $$ which follows from $\Delta$ being a Hecke eigenform appreciated weight 12.
References
Ramanujan function.
Encyclopedia of Mathematics. URL: ?title=Ramanujan_function&oldid=52820