M.ram Murty V.kumar Murty

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-^)-!:.,^ It is conjectured by ATKIN and SERRE (6, equation 4. 11 A;] that for any c>0. l^)!^^ In 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 Research of the first two author ^a;» partial!) supported b\ NSERC grants L'0237 and lTn7? respecmelv M R Ml im. Dt'fhirtmt'ni «»f Maihcmulu \. MtGtl! I niu'r\n\. Mimimil. CumiUii V K. Ml R T ^ . Department »*/ Mathematics Cimcnruta (.''mrm/fi. Montreal. Canada T N SHour^. .Srh«N»/ of Mathemaiu\. Tata InMnuie 0, such that for all positive integers n for which T (n) is odd^ we have \x(n)\^(\ognY. It follows from the theorem that for an odd integer a, the equation (1) T(n)==a has only finitely many solutions. As •:(?) is even, all integers satisfying (1) are squarefull (i.e. every prime divisor of n appears to at least the second power). We apply the theory of linear forms in logarithms to obtain lower bounds for T (/?'"), p a prime and m ̂ 2, which, in particular gives the theorem. We require several lemmas. LEMMA 1. — T^)^ if and only ifm is odd and i(^)=0. Proof. Write T(/?)=ap+a^, a^/? ̂ \ O^O^TI. Set ^(P)-' 1 , if mis even T (p), if m is odd and ^=exp(27ii7(w-h 1)). Then, as in Ramanujan [4], (2) T^^a;'-^)^-^) -T.WlT^^-^apX^-r^,) ^(P)^^^^^?^^^^^' If the r-th factor is zero, 4cos(Jl^/(w+l))=74-r +2=T(^) /p l l , is both an algebraic integer and a rational number. Thus it is a rational integer and so must be one of 1, 2 or 3. But none of T^)—/?, ^(p)-!?, x^)-^ can be zero, since i (2)=-24^±2 6 and T (3) = 252 ̂ ± 3. Thus T (p^ = 0 if and only if y^ (p) = 0. The next three lemmas depend on the theory of linear forms in logarithms. They are stronger than needed for the proof of the theorem. They may be of independent interest. TOME 1 1 5 1987 N 3 ODD VALL bS Oh THE RAMANUJAN T-FUNCTION 393 LEMMA 2. — There is an effectively computable absolute constant C, >0 such that for all m ̂ 2, we have l^"')! ^ IVm^)!^ 1 1 ' Proo/. — Suppose that m is odd. If x(p)=0, there is nothing to prove. If x(/?)=40, then we see from (2) that T (/?'") ̂ 0. Then T^^Wl^la^-^'!!^-^;-