John A. Ewell
Abstract
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Several formulas for Ramanujan’s function τ \tau , defined by \[ x ∏ 1 ∞ ( 1 − x n ) 24 = ∑ 1 ∞ τ ( n ) x n ( | x | > 1 ) , x \prod _1^{\infty }(1-x^n)^{24}=\sum _1^{\infty } \tau (n) x^n \quad (\vert x \vert > 1), \] are presented. We also present a congruence modulo 3 for some of the function values.
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Several formulas for Ramanujan’s function τ \tau , defined by \[ x ∏ 1 ∞ ( 1 − x n ) 24 = ∑ 1 ∞ τ ( n ) x n ( | x | > 1 ) , x \prod _1^{\infty }(1-x^n)^{24}=\sum _1^{\infty } \tau (n) x^n \quad (\vert x \vert > 1), \] are presented. We also present a congruence modulo 3 for some of the function values.
Key concepts: Ramanujan's sum, Modulo, Ramanujan tau function, Mathematics, Congruence (geometry), Combinatorics, Ramanujan theta function, Function (biology)