On some Formulae for Ramanujan's tau Function

Some formulae of Niebur and Lanphier are derived in an elementary manner from previously known formulae. A new congruence formula for $\tau(p)$ modulo $p$ is derived as a consequence. We use this congruence to numerically investigate the order of $\tau(p)$ modulo $p$.

On some Formulae for Ramanujan's tau Function Sobre algunas fórmulas para la función tau de Ramanujan LUIS H. GALLARDO1

1University of Brest, Brest, France. Email: Luis.Gallardo@univ-brest.fr 

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Abstract

Some formulae of Niebur and Lanphier are derived in an elementary manner from previously known formulae. A new congruence formula for τ(p) modulop is derived as a consequence. We use this congruence to numerically investigate the order of τ(p) modulo p.

Key words: Ramanujan's tau formulae, Congruences.

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2000 Mathematics Subject Classification: 11A25, 11A07.

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Resumen

Obtenemos algunas fórmulas de Niebur y Lanphier de manera elemental a partir de formulas conocidas. Deducimos una nueva fórmula para la congruencia τ(p) modulo p. Utilizamos esa fórmula para estudiar numéricamente el orden multiplicativo de τ(p) modulo p.

Palabras clave: Fórmulas para la función tau de Ramanujan, congruencias.

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Texto completo disponible en PDF

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References

[1] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edn, Springer-Verlag, New York, United States, 1990.

[2] I. Cherednik, `A note on Artin's Constant´, ArXiv math. NT 0810.2325v3, 2008.

[3] S. Chowla, `Note on a Certain Arithmetical Sum´, Proc. Nat. Inst. Sci. India 13, 5 (1947), 1-1.

[4] L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea Publishing Company, New York, United States, 1992.

[5] J. W. L. Glaisher, `On the Square of the Series in which the Coefficients are the Sum of the Divisors of the Exponents´, Messenger of Math. 14, (1884), 156-163.

[6] J. W. L. Glaisher, `Expressions for the First Five Powers of the Series in which the Coefficients are the Sums of the Divisors of the Exponents´, Messenger of Math. 15, (1885), 33-36.

[7] F. Q. Gouvêa, `Non-ordinary Primes: A Story´, Experiment. Math. 6, 3 (1997), 195-205.

[8] J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Williams, `Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions´, Number theory for the Millenium II, (2002), 229-274.

[9] D. B. Lahiri, `On Ramanujan's function τ(n) and the divisor function σk(n)-I´, Bull. Calcutta Math. Soc. 38, (1946), 193-206.

[10] D. Lanphier, `Maass Operators and van der Pol-type Identities for Ramanujan's tau Function´, Acta Arith.113, 2 (2004), 157-167.

[11] D. H. Lehmer, Some Functions of Ramanujan in Selected Papers of D. H. Lehmer, Vol. II, Charles Babbage Research Centre, Box 370, St. Pierre, Manitoba, Canada, 1981. Reprinted from Math. Student, Vol. 27 (1959), pp. 105-116

[12] D. Niebur, `A Formula for Ramanujan's τ-function´, Illinois J. Math. 19, (1975), 448-449.

[13] M. Papanikolas, `A Formula and a Congruence for Ramanujan's τ-function´, Proc. Amer. Math. Soc. 134, 2 (2006), 333-341.

[14] B. v. d. Pol, `On a Non-Linear Partial Differential Equation Satisfied by the Logarithm of the Jacobian Theta-Functions, with Arithmetical Applications. I and II´, Indag. Math. 13, (1951), 261-284.

[15] S. Ramanujan, Collected Papers of Srinivasa Ramanujan, `On Certain Arithmetical Functions´, Cambridge at The University Press, chapter 18, 1927, p. 136-162. Reprinted from Transactions of the Cambridge Philosophical Society, XXII, No. 9, 1916, pp. 159-184

[16] H. L. Resnikoff, `On Differential Operators and Automorphic Forms´, Trans. Amer. Math. Soc. 124, 334--346 (1968).

[17] J. P. Serre, `Une interprétation des congruences relatives à la fonction τ de Ramanujan´, Séminaire Delange-Pisot-Poitou: 1967/68, Théorie des Nombres 1, 14 (1969), 1-17.

[18] J. P. Serre, `An Interpretation of some Congruences Concerning Ramanujan's τ-function´, Published on line at http://www.rzuser.uni-heidelberg.de/hb3/serre.ps, 1997.

[19] N. J. A. Sloane, `The On-Line Encyclopedia of Integers Sequences´, Published on line at http://www.research.att.com/sequences, 2007.

[20] A. Straub, `Ramanujan's τ-function. With a Focus on Congruences´, Published on line at http://arminstraub.com, 2007. pp. 1-13

[21] J. Touchard, `On Prime Numbers and Perfect Numbers´, Scripta Math. 19, (1953), 35-39.

(Recibido en agosto de 2009. Aceptado en octubre de 2010)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv44n2a03, 
    AUTHOR  = {Gallardo, Luis H.}, 
    TITLE   = {{On some Formulae for Ramanujan's tau Function}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2010}, 
    volume  = {44}, 
    number  = {2}, 
    pages   = {103-112} 
}