Ramanujan’s Notebooks

Abstract

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On the modular behaviour of the infinite product (1−x)(1−xq)(1−xqZhang1 2011Comptes Rendus. Mathématique2000View full textAdd to dashboardCite

“…Losque q tend vers la valeur unité, q * devient exponentiellement petit ; ainsi la série (ι q x; q * ) ∞ échappe-t-elle à toute représentation en termes de série entière, convergente ou semi-convergente, de la variable log q. Dans le Notebooks de Ramanujan, on trouve au moins deux formules consacrées au développement du produit (x; q) ∞ au moyen du dilogarithme et d'une série entière de log q ; elles donnent lieu aux Entry 6, p. 265 et Entry 6 , p. 268 de [5]. Entry 6 est interprétée dans [5] comme une formule asymptotique de log(x; q) ∞ alors que le manuscrit de Ramanujan utilise constamment le symbole « = », la plupart de ses séries entières rencontrées étant « semi-convergentes ».…”

Values of Dirichlet L-functions, Gauss sums and trigonometric sums

“…As Berndt and Zaharescu remarked, their method is very flexible and can be applied to a plenty of sums of similar type. They also mention that the main inspiration of their work comes from identities discovered by Berndt and Zhang [7] in the course of proving two difficult theta function identities from Ramanujan's notebooks [18] (see also [4]). Precisely, the identities sin( 2π 7 ) sin 2 ( 3π 7 )…”

Partition identities arising from Ramanujan’s formulas for multipliers

“…Ramanujan recorded many modular equations in his notebooks [31] and the lost notebook [32]. For proofs of the modular equations of Ramanujan, we refer to the books by Berndt [7][8][9] and by Andrews and Berndt [1,Chap. 17].…”