Nonvanishing of HeckeL-Functions Associated to Cusp Forms inside the Critical Strip
✍ Scribed by Winfried Kohnen
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 274 KB
- Volume
- 67
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Let f be a non-zero cuspidal Hecke eigenform of integral weight k on the full modular group SL 2 (Z) and denote by L*( f, s) (s # C) the associated Hecke L-function completed with its natural 1-factor. As is well-known, zeroes of L*( f, s) can occur only inside the critical strip (k&1)Â2< Re(s)<(k+1)Â2, and according to the generalized Riemann hypothesis they should all lie on the line Re(s)=kÂ2.
While at present a proof of the Riemann hypothesis seems to be out of reach, it turns out to be rather easy to show nonvanishing results for L-functions on the average. In fact, in the present paper we shall prove that, given a real number t 0 and a positive real number =, for all k large enough the sum of the functions L*( f, s) with f running over a basis of (properly normalized) Hecke eigenforms of weight k does not vanish on the line segments Im(s)=t 0 , (k&1)Â2<Re(s)<(kÂ2)&=, (kÂ2)+=<Re(s)< (k+1)Â2. For the proof we consider the cusp forms dual w.r.t. the Petersson scalar product of the values L*( f, s); these functions are generalizations of the period functions studied in [3]. We compute their Fourier expansion and estimate the first Fourier coefficient in an appropriate way. The result then will turn out.
K. Murty kindly informed the author that A. Akbary in his thesis independently proved by a different method using an approximate functional equation for an average sum of L-functions results similar to those presented here.
For a related result on Rankin convolutions we refer the reader to [4] (approximate functional equations are also used there).