The nonvanishing of almost prime twists of modular L-functions
\(L\)-functions are special types of Dirichlet series which often hold fundamental arithmetic information. Hence, they are among the most important objects in analytic number theory. In this talk, we consider the so-called Hecke \(L\)-function \(L(s,f,X_d)\) associated to a given normalized holomorphic newform \(f\) twisted by the Kronecker symbol \(X_d\). It is well known that the twisted \(L(s,f,X_d)\) converges absolutely for \(Re(s)>1\) and admits a functional equation which extends it analytically to the whole complex plane. The value of \(L(s,f,X_d)\) at \(s=1/2\) is of special interest. For instance, if the form \(f\) parametrizes an elliptic curve \(E\), then the Birch-Swinnerton-Dyer conjecture asserts that the rank of \(E\) twisted by a fundamental discriminant \(d\) is precisely the order of vanishing of \(L(s,f,X_d)\) at \(s=1/2\).
In this project, we fix a holomorphic newform \(f\) of weight at least \(2\), level \(N\) with trivial nebentype and consider the family of twisted \(L\)-functions \(L(s,f,X_d)\) where \(d\) is any odd fundamental discriminant with \((d,N)=1\). Using an adaptation of a method by Iwaniec, we prove that there are infinitely many fundamental discriminants \(d\) such that \(L(1/2,f,X_d)\) is not equal to \(0\). In addition, following an idea outlined by Hoffstein and Luo, using combinatorial sieve, we prove that the same holds for infinitely many almost-prime fundamental discriminants \(d\) with at most \(84\) prime factors. Further improvement of this result, which relies on properties of some multiple-Dirichlet series, are also discussed. Under some assumptions on certain weight factors, it is possible to reduce the number \(84\) to just \(4\). In this presentation, I will give an outline of each of these techniques and discuss possible future projects.