On the density function for the value-distribution of automorphic $L$-functions

On the value-distribution of the difference between logarithms of two symmetric power L-functions

“…Murty [34] for the full modular case, and M. R. Murty and V. K. Murty [35] for any level N ). In [28], we observed geometric behavior of the corresponding curves and proved Lemma 5.4 ([28]). If p n ∈ P f (ε) and n is sufficiently large, then…”

“…The basic structure of the proof of Theorem 5.1 in [28], which we briefly outline here, is along the line similar to [24]. Actually in [28], we are working in more general situation, that is in the class M . Let ϕ ∈ M .…”

“…The method in [24] is to use Lévy's convergence theorem, again in probability theory. This method can also be applied to general ϕ ∈ M , which is pointed out in [25] and a sketch of the argument in the general case is described in [28].…”

“…Automorphic L-functions are typical examples for which the corresponding curves are not always convex, so it is important how to generalize the part (ii) of Theorem 1.1 to the case of automorphic L-functions L(f, s). This has been done in [28].…”

“…where z n (θ n ; ϕ) is defined by (2.6) and z, w = ℜzℜw + ℑzℑw. In [28], we prove the following Lemma 5.3 ([28]). If there are at least five n's for which…”