[go: up one dir, main page]
TOPICS
Search

Tau Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
TauFunction

A function tau(n) related to the divisor function sigma_k(n), also sometimes called Ramanujan's tau function. It is defined via the Fourier series of the modular discriminant Delta(tau) for tau in H, where H is the upper half-plane, by

Delta(tau)=(2pi)^(12)sum_(n=1)^inftytau(n)e^(2piintau)
(1)

(Apostol 1997, p. 20). The tau function is also given by the Cauchy product

tau(n)=8000{(sigma_3 degreessigma_3) degreessigma_3}(n)-147(sigma_5 degreessigma_5)(n)
(2)
=(65)/(756)sigma_(11)(n)+(691)/(756)sigma_5(n)-(691)/3sum_(k=1)^(n-1)sigma_5(k)sigma_5(n-k),
(3)

where sigma_k(n) is the divisor function (Apostol 1997, pp. 24 and 140), sigma_3(0)=1/240, and sigma_5(0)=-1/504.

The tau function has generating function

G(x)=sum_(n=1)^(infty)tau(n)x^n
(4)
=xproduct_(n=1)^(infty)(1-x^n)^(24)
(5)
=x(x)_infty^(24)
(6)
=x-24x^2+252x^3-1472x^4+4830x^5-6048x^6+...
(7)
=x(1-3x+5x^3-7x^6+...)^8,
(8)

where (q)_infty=(q;q)_infty is a q-Pochhammer symbol. The first few values are 1, -24, 252, -1472, 4830, ... (OEIS A000594). The tau function is given by the Wolfram Language function RamanujanTau[n].

The series

f(s)=sum_(n=1)^infty(tau(n))/(n^s),
(9)

is known as the tau Dirichlet series.

Lehmer (1947) conjectured that tau(n)!=0 for all n, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for n<214928639999 (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of n for which this condition holds.

nreference
3316799Lehmer (1947)
214928639999Lehmer (1949)
10^(15)Serre (1973, p. 98), Serre (1985)
1213229187071998Jennings (1993)
22689242781695999Jordan and Kelly (1999)
22798241520242687999Bosman (2007)

Ramanujan gave the computationally efficient triangular recurrence formula

(n-1)tau(n)=sum_(m=1)^(b_n)(-1)^(m+1)(2m+1)×[n-1-9/2m(m+1)]tau(n-1/2m(m+1)),
(10)

where

b_n=1/2(sqrt(8n+1)-1)
(11)

(Lehmer 1943; Jordan and Kelly 1999), which can be used recursively with the formula

tau(p^n)=sum_(j=0)^(|_n/2_|)(-1)^j(n-j; n-2j)p^(11j)[tau(p)]^(n-2j)
(12)

(Gandhi 1961, Jordan and Kelly 1999).

Ewell (1999) gave the beautiful formulas

tau(4n+2)=-3sum_(k=1)^(2n+1)2^(3b(2k))sigma_3(Od(2k))
(13)
×sum_(j=0)^(4n-2k+2)(-1)^jr_8(4n+2-2k-j)r_8(j)
(14)
sum_(k=1)^(n)2^(3b(2k))sigma_3(Od(2k))
(15)
×sum_(j=0)^(2n+1-2k)(-1)^jr_8(2n+1-2k-j)r_8(j)=0
(16)
tau(4m)=-2^(11)tau(m)-3sum_(k=1)^(2m)2^(3b(2k))sigma_3(Od(2k))
(17)
×sum_(j=0)^(4m-2k)(-1)^jr_8(4m-2k-j)r_8(j)
(18)
tau(2n+1)=sum_(k=1)^(2n+1)2^(3[b(2k)-1])sigma_3(Od(2k))
(19)
×sum_(j=0)^(2n+2-2k)(-1)^jr_8(3n+2-2k-j)r_8(j),
(20)

where b(n) is the exponent of the exact power of 2 dividing n, Od(n) is the odd part of n, sigma_k(n) is the divisor function of n, and r_k(n) is the sum of squares function.

For prime p,

tau(p^(n+1))=tau(p)tau(p^n)-p^(11)tau(p^(n-1))
(21)

for n>=1, and

tau(p^alphan)=tau(p)tau(p^(alpha-1)n)-p^(11)tau(p^(alpha-2)n)
(22)

for alpha>=2 and (n,p)=1 (Mordell 1917; Apostol 1997, p. 92).

Ramanujan conjectured and Mordell (1917) proved that if (n,n^')=1, then

tau(nn^')=tau(n)tau(n^')
(23)

(Hardy 1999, p. 161). More generally,

tau(n)tau(n^')=sum_(d|(n,n^'))d^(11)tau((nn^')/(d^2)),
(24)

which reduces to the first form if (n,n^')=1 (Mordell 1917; Apostol 1997, p. 93).

Ramanujan (1920) showed that

tau(2n)=0 (mod 2)
(25)
tau(3n)=0 (mod 3)
(26)
tau(5n)=0 (mod 5)
(27)

(Darling 1921; Wilton 1930),

tau(7n+m)=0 (mod 7)
(28)

for m=0 or one the quadratic non-residues of 7, i.e., 3, 5, 6, and

tau(23n+m)=0 (mod 23)
(29)

for one the quadratic non-residues of 23, i.e., 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 (Mordell 1922; Wilton 1930). Ewell (1999) showed that

tau(4n)=tau(n) (mod 3).
(30)

Ramanujan conjectured and Watson proved that tau(n) is divisible by 691 for almost all n, specifically

tau(n)=sigma_(11)(n) (mod 691),
(31)

where sigma_k(n) is the divisor function (Wilton 1930; Apostol 1997, pp. 93 and 140; Jordan and Kelly 1999), and 691 is the numerator of the Bernoulli number B_(12).

Additional congruences include

tau(n)=sigma_(11)(n) (mod 2^8) for n odd
(32)
tau(n)=n^2sigma_7(n) (mod 3^3)
(33)
tau(n)=nsigma_9(n) (mod 5^2)
(34)
tau(n)=nsigma_3(n) (mod 7)
(35)
tau(n)=sigma_(11)(n) (mod 691)
(36)
tau(n)={sigma_(11)(n) (mod 2^(11)) if n=1 (mod 8); 1217sigma_(11)(n) (mod 2^(13)) if n=3 (mod 8); 1537sigma_(11)(n) (mod 2^(12)) if n=5 (mod 8); 705sigma_(11)(n) (mod 2^(14)) if n=7 (mod 8)
(37)
tau(n)=n^(-610)sigma_(1231)(n){ (mod 3^6) if n=1 (mod 3); (mod 3^7) if n=2 (mod 3)
(38)
tau(n)=n^(-30)sigma_(71)(n) if GCD(n,5)=1
(39)
tau(n)=nsigma_9(n){ (mod 7) if n=0, 1, 2, or 4 (mod 7); (mod 7^2) if n=3, 5, or 6 (mod 7)
(40)
tau(p)={0 (mod 23) if (p/23)=-1; 2 (mod 23) if p=u^2+23v^2 with u!=0, v; -1 (mod 23) for other p!=23,
(41)

where sigma_k(n) is the divisor function (Swinnerton-Dyer 1988, Jordan and Kelly 1999).

tau(n) is almost always divisible by 2^5·3^3·5^2·7^2·23·691 according to Ramanujan. In fact, Serre has shown that tau(n) is almost always divisible by any integer (Andrews et al. 1988).

The summatory tau function is given by

T(n)=sum^'_(n<=x)tau(n).
(42)

Here, the prime indicates that when x is an integer, the last term tau(x) should be replaced by 1/2tau(x).

See also

Dedekind Eta Function, j-Function, Leech Lattice, Ore's Conjecture, Partition Function P, Tau Conjecture, Tau Dirichlet Series, Tau Function Prime

Explore with Wolfram|Alpha

References

Andrews, G. E.; Berndt, B. C.; and Rankin, R. A. (Eds.). Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987 New York: Academic Press, 1988.Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, 1997.Charles, C. D. "Computing the Ramanujan Tau Function." http://www.cs.wisc.edu/~cdx/.Darling, H. B. C. Proc. London Math. Soc. 19, 350-372, 1921.Ewell, J. A. "New Representations of Ramanujan's Tau Function." Proc. Amer. Math. Soc. 128, 723-726, 1999.Gandhi, J. M. "The Nonvanishing of Ramanujan's tau-Function." Amer. Math. Monthly 68, 757-760, 1961.Hardy, G. H. "Ramanujan's Function tau(n)." Ch. 10 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 63 and 161-185, 1999.Jennings, D. Ph.D. thesis. Southampton, 1993.Jordan, B. and Kelly, B. III. "The Vanishing of the Ramanujan Tau Function." Preprint, 12 Mar 1999.Keiper, J. "On the Zeros of the Ramanujan tau-Dirichlet Series in the Critical Strip." Math. Comput. 65, 1613-1619, 1996.LeVeque, W. J. §F35 in Reviews in Number Theory 1940-1972. Providence, RI: Amer. Math. Soc., 1974.Lehmer, D. H. "Ramanujan's Function tau(n)." Duke Math. J. 10, 483-492, 1943.Lehmer, D. H. "The Vanishing of Ramanujan's Function tau(n)." Duke Math. J. 14, 429-433, 1947.Moreno, C. J. "A Necessary and Sufficient Condition for the Riemann Hypothesis for Ramanujan's Zeta Function." Illinois J. Math. 18, 107-114, 1974.Mordell, L. J. "On Mr. Ramanujan's Empirical Expansions of Modular Functions." Proc. Cambridge Phil. Soc. 19, 117-124, 1917.Mordell, L. J. "Note on Certain Modular Relations Considered by Messrs Ramanujan, Darling, and Rogers." Proc. London Math. Soc. 20, 408-416, 1922.Ramanujan, S. Proc. London Math. Soc. 18, 1920.Ramanujan, S. "Congruence Properties of Partitions." Math. Z. 9, 147-153, 1921.Serre, J.-P. A Course in Arithmetic. New York: Springer-Verlag, 1973.Serre, J.-P. "Sur la Lacunatité des Puissances de eta." Glasgow Math. J. 27, 203-221, 1985.Sivaramakrishnan, R. Classical Theory of Arithmetic Functions. New York: Dekker, pp. 275-278, 1989.Sloane, N. J. A. Sequence A000594/M5153 in "The On-Line Encyclopedia of Integer Sequences."Spira, R. "Calculation of the Ramanujan Tau-Dirichlet Series." Math. Comput. 27, 379-385, 1973.Stanley, G. K. "Two Assertions Made by Ramanujan." J. London Math. Soc. 3, 232-237, 1928.Stanley, G. K. Corrigendum to "Two Assertions Made by Ramanujan." J. London Math. Soc. 4, 32, 1929.Swinnerton-Dyer, H. P. F. "Congruence Properties of tau(n)." In Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987 (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). New York: Academic Press, 1988.Watson, G. N. "Über Ramanujansche Kongruenzeigenschaften der Zerfällungsanzahlen." Math. Z. 39, 712-731, 1935.Wilton, J. R. "Congruence Properties of Ramanujan's Function tau(n)." Proc. London Math. Soc. 31, 1-17, 1930.Yoshida, H. "On Calculations of Zeros of L-Functions Related with Ramanujan's Discriminant Function on the Critical Line." J. Ramanujan Math. Soc. 3, 87-95, 1988.

Referenced on Wolfram|Alpha

Tau Function

Cite this as:

Weisstein, Eric W. "Tau Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TauFunction.html

Subject classifications