Tau Function

The tau function , also called Ramanujan's tau function, is defined for positive integers and is related to the divisor function . It is defined via the Fourier series of the modular discriminant for , where is the upper half-plane, by

(1)

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

		$8000{(sigma_3 degreessigma_3) degreessigma_3}(n)-147(sigma_5 degreessigma_5)(n)$	(2)
			(3)

where is the divisor function (Apostol 1997, pp. 24 and 140), , and .

The tau function has generating function

			(4)
			(5)
			(6)
			(7)
			(8)

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

The series

(9)

for is known as the tau Dirichlet series.

Lehmer (1947) conjectured that for all positive integers , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for (Apostol 1997, p. 22). The following table summarizes historical bounds for which holds for all positive integers .

	reference
3316799	Lehmer (1947)
214928639999	Lehmer (1949)
	Serre (1973, p. 98), Serre (1985)
1213229187071998	Jennings (1993)
22689242781695999	Jordan and Kelly (1999)
22798241520242687999	Bosman (2007, p. 3)

Ramanujan gave the computationally efficient triangular recurrence relation

(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)

for , with and , where is the floor function (Lehmer 1943, Jordan and Kelly 1999). It can be used with the formula

(11)

for prime and nonnegative integers (Gandhi 1961, Jordan and Kelly 1999).

Ewell (1999) gave the beautiful formulas

	(12)
	(13)
	(14)
	(15)
	(16)
	(17)
	(18)
	(19)

where is a nonnegative integer and is a positive integer in these formulas, , and is the exponent of the exact power of 2 dividing the positive integer , is the odd part of , is the divisor function of , and is the sum of squares function.

For prime ,

(20)

for , and

(21)

for integers and (Mordell 1917; Apostol 1997, p. 92).

Ramanujan conjectured and Mordell (1917) proved that if , then

(22)

(Hardy 1999, p. 161). More generally,

(23)

which reduces to the first form if (Mordell 1917; Apostol 1997, p. 93).

For integers for which the displayed arguments are positive, Ramanujan (1920) showed that

	(24)
	(25)
	(26)

(Darling 1921, Wilton 1930),

(27)

for or one of the quadratic nonresidues of 7, i.e., 3, 5, 6, and

(28)

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

(29)

Ramanujan conjectured and Watson proved that is divisible by 691 for almost all , specifically

(30)

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

Additional congruences for positive integers include

			(31)
			(32)
			(33)
			(34)
			(35)
		${sigma_(11)(n) (mod 2^(11)) if n=1 (mod 8); 1217sigma_(11)(n) (mod 2^(13)) if n=3 (mod 8); 1537sigma_(11)(n) (mod 2^(12)) if n=5 (mod 8); 705sigma_(11)(n) (mod 2^(14)) if n=7 (mod 8)$	(36)
		$n^(-610)sigma_(1231)(n){ (mod 3^6) if n=1 (mod 3); (mod 3^7) if n=2 (mod 3)$	(37)
			(38)
		$nsigma_9(n){ (mod 7) if n=0, 1, 2, or 4 (mod 7); (mod 7^2) if n=3, 5, or 6 (mod 7)$	(39)
		${0 (mod 23) if (p/23)=-1; 1 (mod 23) if p=23; 2 (mod 23) if p=u^2+23v^2 with u!=0; -1 (mod 23) for other p!=23,$	(40)

where is a prime number, and are integers, is the Legendre symbol, and is the divisor function (Lehmer 1947, p. 432; Serre 1968, p. 14-03; Swinnerton-Dyer 1988, Jordan and Kelly 1999). Negative powers of denote powers of its modular inverse modulo the indicated modulus.

is almost always divisible by according to Ramanujan. In fact, for every fixed positive integer , is divisible by for almost all positive integers (Serre 1976).

The summatory function of is defined for positive real numbers by

(41)

Here, the prime indicates that when is an integer, the last term should be replaced by .

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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.Bosman, J. "On the Computation of Galois Representations Associated to Level One Modular Forms." 5 Oct 2007. https://arxiv.org/abs/0710.1237.Charles, C. D. "Computing the Ramanujan Tau Function." https://pages.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. https://doi.org/10.1090/S0002-9939-99-05289-2.Gandhi, J. M. "The Nonvanishing of Ramanujan's

-Function." Amer. Math. Monthly 68, 757-760, 1961.Hardy, G. H. "Ramanujan's Function

." 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. "An Investigation into Modular Forms,

-Series, Partitions and Related Applications." PhD thesis. Southampton, England: University of Southampton, January 1993. https://eprints.soton.ac.uk/462198/.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

-Dirichlet Series in the Critical Strip." Math. Comput. 65, 1613-1619, 1996.Lehmer, D. H. "Ramanujan's Function

." Duke Math. J. 10, 483-492, 1943.Lehmer, D. H. "The Vanishing of Ramanujan's Function

." Duke Math. J. 14, 429-433, 1947.LeVeque, W. J. §F35 in Reviews in Number Theory 1940-1972. Providence, RI: Amer. Math. Soc., 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.Moreno, C. J. "A Necessary and Sufficient Condition for the Riemann Hypothesis for Ramanujan's Zeta Function." Illinois J. Math. 18, 107-114, 1974.Ramanujan, S. Proc. London Math. Soc. 18, 1920.Ramanujan, S. "Congruence Properties of Partitions." Math. Z. 9, 147-153, 1921.Serre, J.-P. "Une interprétation des congruences relatives à la fonction

de Ramanujan." Séminaire Delange-Pisot-Poitou. Théorie des nombres 9, 14-01-14-17, 1968. https://www.numdam.org/item/SDPP_1967-1968__9_1_A13_0/.Serre, J.-P. A Course in Arithmetic. New York: Springer-Verlag, 1973.Serre, J.-P. "Divisibilité de certaines fonctions arithmétiques." Enseign. Math. 22, 227-260, 1976. https://doi.org/10.5169/seals-48187.Serre, J.-P. "Sur la Lacunatité des Puissances de

." 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

." 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

." 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.

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Tau Function

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

Tau Function

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