From the text: This note should be seen as an addendum to the work of E. Kowalski and the second author [Duke Math. J. 100, 503–542 (1999; Zbl 1161.11359)] and deals with the problem of bounding, unconditionally, the order of vanishing at the critical point in a family of \(L\)-functions. This problem is illustrated in the particular case of \(L\)-functions of weight-2 primitive modular forms of prime level.
Recall the notation from that paper: For q a prime number, let \(S_2(q)^*\) be the set of primitive forms of weight 2 and level \(q\), normalized so that their first Fourier coefficient is 1; for \(f (z) =\sum_{n\geq 1} \lambda_f(n)n^{1/2}e(nz) \in S_2(q)^*\), \(\lambda_f (1) = 1\), let \(L(f, s) :=\sum_{n\geq 1}\lambda_f (n)n^{-s}\), the associated (normalized) \(L\)-function; it admits analytic continuation to \(\mathbb C\) with a functional equation relating \(L(f, s)\) to \(L(f,1-s),\) and we call \(r_f := \text{ord}_{s=1/2} L(f, s)\) the analytic rank of \(f\) .
In [Kowalski-Michel (loc. cit.)] the following was proved: There exists an absolute constant \(C_1 \geq 0\) such that, for all \(q\) prime, \(\sum_{f \in S_2(q)^*} r_f \leq C_1\,| S_2(q)^*|.\)
After that, much progress has been made concerning this question. In particular, in [E. Kowalski, P. Michel and J. M. VanderKam, J. Reine Angew. Math. 526, 1–34 (2000; Zbl 1020.11033)], a sharp explicit value was given for the constant \(C_1\) (\(C_1 = 1.1891\) for \(q\) large enough). In the course of the proof, a uniform bound for the square of the ranks was obtained: \(\sum_{f \in S_2(q)^*}r_f ^2 \leq C_2\,| S_2(q)^*|.\)
However, the latter improvement used only a slight variant of the methods of Kowalski-Michel (loc. cit.). In fact, it is possible to pursue this idea further and it turns out that much more is true; this is the subject of the present note.
Consider a finite probability space \((\Omega,\mu)\), where \(\mu(\omega) > 0\) for every \(\omega\in \Omega\). For each \(\omega\in \Omega\), suppose given a function \(h_\omega(s)\) which is holomorphic in the half plane \(\text{Re}(s) \geq 0\). Moreover assume the following hypothesis on the variance of the function \(h_\omega(s)-1\):
Hypothesis: For some \(B,C > 0\), \(M > 2\), we have the bound
\[ \sum_{\omega\in \Omega}|h_\omega(\sigma + it) - 1|^2\mu(\omega) \leq C(1 + |t|)^BM^{-\sigma} \] uniformly for \(\sigma\geq (2 \log M)^{-1}\).
Note that in view of this hypothesis and the fact that \(\mu(\omega) > 0\), we have
\[ h_{\omega}(s) = 1 + O_{\omega}((1 + |t|)^{B/2}M^{-\sigma/2}) \tag{(*)} \]
for each \(h_{\omega}\). Thus \(h_{\omega}(s)\) is nonvanishing for sufficiently large \(\sigma\). For any \(\alpha\geq 0\) and \(t_1 < t_2 \in \mathbb R\), we may therefore define \(N(\omega, \alpha, t_1, t_2)\) to be the number of zeros \(\rho\) of \(h_{\omega}(s)\) such that \(\text{Re}(\rho)\geq \alpha\), \(t_1\leq \text{Im}(\rho)\leq t_2\). Clearly \(N(\omega, \alpha, t_1, t_2)\) is finite. Our general result gives an upper bound for the \(2k\)-th power of \(N(\omega, \alpha, t_1, t_2)\) on average:
Theorem: With the above notations, assume that the above Hypothesis is satisfied. Then for all \(k\geq 1\), for all \(\alpha\geq (\log M)^{-1}\), and all \(t_1 < t_2,\) we have
\[ \sum_{\omega\in \Omega} N(\omega, \alpha, t_1, t_2)^{2k}\mu(\omega)\ll C(k!)^2\left(48 \frac{k}{\alpha \log M}\right)^{2k} \left(1+|t|+\frac{16k}{\log M}\right)^BM^{-\alpha/2}(1+(t_2-t_1) \log M) \]
where we have set \(|t| := \max(|t_1|, |t_2|)\). The constant involved in the Vinogradov symbol is absolute.