Distribution of values of L -functions at the edge of the critical strip

Abstract

We prove several results on the distribution of values of L‐functions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power L‐functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these L‐functions) at s = 1, functions in the Selberg class (in the height aspect) and quadratic twists of a fixed GL(m)/ℚ‐automorphic cusp form at s = 1.

Distribution of values of symmetric power L-functions at the edge of the critical strip

“…uniformly for all prime numbers N and t log 2 N − log 3 N − 2 log 4 N. We note that the domain of validity of t is slightly lager than that of (1.32) but the error term is slightly weaker than that of (1.32). By refining Lamzouri's method [12], we can prove the following result.…”

“…In this section, we will refine the argument of Lamzouri [12] and apply a little more tricks from [15] to proof Theorem 3. We only consider the case of sign −, and the other case can be treated in the same way.…”

Large Moments and Extreme Values of Class Numbers of Indefinite Binary Quadratic Forms

“…Later in [12], Raulf generalized her method to obtain an asymptotic formula for the k-th moment of h(d), for any fixed natural number k. We improve on this result, by obtaining an asymptotic formula for the k-th moment of h(d) over positive discriminants d with ε d ≤ x, uniformly for all real numbers k in the range 0 < k ≤ (log x) 1−o (1) . Our approach is different, and relies on the methods of Granville and Soundararajan [4] and the author ( [7] and [8]) for computing large moments of L(1, χ d ). Here and throughout we let log j be the j-fold iterated logarithm; that is, log 2 = log log, log 3 = log log log and so on.…”

Value-distribution of cubic Hecke L-functions

“…(Note that since c P Z c , the integrand has only a simple pole at u " 0. )Employing the above lemma in(20) for σ ď 1 yields(22) ‹ ÿ cPC exp piyL c pσqq expp´Npcq{Yq " pIq´pIIq`pIIIq`OpY δ q, Npaq σ Npbq σ expˆ´N Npaq σ Npbq σ expˆ´N exp piyL c pσ`uqq ΓpuqX u du¸expˆ´N pcq Y˙.…”