Ramanujan's Tau Function

The Ramanujan tau function, studied by , is the function $\tau : \mathbb \rarr\mathbb$ defined by the following identity:
:$\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z),$
where with , $\phi$ is the Euler function, is the Dedekind eta function, and the function is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write $\Delta/(2\pi)^$ instead of $\Delta$). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in .

Values

The first few values of the tau function are given in the following table :

Ramanujan's conjectures

observed, but did not prove, the following three properties of :

* if (meaning that is a multiplicative function)
* for prime and .
* for all primes .

The first two properties were proved by and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by applying them to a Kuga-Sato variety).

Congruences for the tau function

For and , define as the sum of the th powers of the divisors of . The tau function satisfies several congruence relations; many of them can be expressed in terms of . Here are some:Page 4 of
#$\tau(n)\equiv\sigma_(n)\ \bmod\ 2^\textn\equiv 1\ \bmod\ 8$Due to
#$\tau(n)\equiv 1217 \sigma_(n)\ \bmod\ 2^\text n\equiv 3\ \bmod\ 8$
#$\tau(n)\equiv 1537 \sigma_(n)\ \bmod\ 2^\textn\equiv 5\ \bmod\ 8$
#$\tau(n)\equiv 705 \sigma_(n)\ \bmod\ 2^\textn\equiv 7\ \bmod\ 8$
#$\tau(n)\equiv n^\sigma_(n)\ \bmod\ 3^\textn\equiv 1\ \bmod\ 3$Due to
#$\tau(n)\equiv n^\sigma_(n)\ \bmod\ 3^\textn\equiv 2\ \bmod\ 3$
#$\tau(n)\equiv n^\sigma_(n)\ \bmod\ 5^\textn\not\equiv 0\ \bmod\ 5$
#$\tau(n)\equiv n\sigma_(n)\ \bmod\ 7\textn\equiv 0,1,2,4\ \bmod\ 7$Due to D. H. Lehmer
#$\tau(n)\equiv n\sigma_(n)\ \bmod\ 7^2\textn\equiv 3,5,6\ \bmod\ 7$
#$\tau(n)\equiv\sigma_(n)\ \bmod\ 691.$

For prime, we have

11. $\tau(p)\equiv 0\ \bmod\ 23\text\left(\frac\right)=-1$
    
12. $\tau(p)\equiv \sigma_(p)\ \bmod\ 23^2\text p\text a^2+23b^2$
    
13. $\tau(p)\equiv -1\ \bmod\ 23\text.$
    

Explicit formula

In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:

:$\tau(n)=n^4\sigma(n)-24\sum_^i^2(35i^2-52in+18n^2)\sigma(i)\sigma(n-i).$

This also shows that the tau function is always an integer.

Conjectures on ''τ''(''n'')

Suppose that is a weight- integer newform and the Fourier coefficients are integers. Consider the problem:
: Given that does not have complex multiplication, do almost all primes have the property that ?
Indeed, most primes should have this property, and hence they are called ''ordinary''. Despite the big advances by Deligne and Serre on Galois representations, which determine for coprime to , it is unclear how to compute . The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes such that , which thus are congruent to 0 modulo . There are no known examples of non-CM with weight greater than 2 for which for infinitely many primes (although it should be true for almost all ). There are also no known examples with for infinitely many . Some researchers had begun to doubt whether for infinitely many . As evidence, many provided Ramanujan's (case of weight 12). The only solutions up to 10¹⁰ to the equation are 2, 3, 5, 7, 2411, and .

conjectured that for all , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for up to (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of for which this condition holds for all .

Ramanujan's ''L''-function

Ramanujan's ''L''-function is defined by
:$L(s)=\sum_\frac$
if $\Re s>6$ and by analytic continuation otherwise. It satisfies the functional equation
:$\frac=\frac,\quad s\notin\mathbb_0^-, \,12-s\notin\mathbb_0^$
and has the Euler product
:$L(s)=\prod_\frac,\quad \Re s>7.$
Ramanujan conjectured that all nontrivial zeros of $L$ have real part equal to $6$.

Notes

References

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*{{Citation
, last=Wilton
, first=J. R.
, title=Congruence properties of Ramanujan's function τ(''n'')
, year=1930
, journal=Proceedings of the London Mathematical Society
, volume=31
, pages=1–10
, doi=10.1112/plms/s2-31.1.1

Modular forms
Multiplicative functions
Srinivasa Ramanujan
Zeta and L-functions