Odd values of the Ramanujan $\tau$-function

Abstract

R^SI'MF -Soil T la fonction de Ramanujan.Nous prouvons qu'il exisle une effeciivement calculable conslante (•>() ahsoluc.lei que si T(n) esi impaire.alors jT(n)j^(logrt)'.Nous ulilisons les resuliats sur les formes lineaires des loganthmes.ABSTRACT -Lei T denote Ramanujan's function.We prove that there exists an effectively computable absolute constant t >0 such that if T(n) is odd.then JT(n)j^(logn)'.We use results on linear forms in logarithms.Ramanujan's T-function is defined by the relation ^n^n-^) 24 -!:., 1 Ît is conjectured by ATKIN and SERRE (6, equation 4. 11 A;] that for any c>0.l^)!^9 2 În particular this implies that for any a, there are only finitely many primes p such that T(/?)=U.In this note.we study a related, though simpler, question.Our main result is the following.(•» Textc re<;u Ie 7 avnl 19Hh

Locations

• Bulletin de la Société mathématique de France - View - PDF