A joint limit theorem for zeta-functions of newforms

In the paper a joint limit theorem for zeta-functions of newforms on the complex plane is proved.

be its Hecke subgroup. Suppose that F (z) is a holomorphic function on the upper half plane Im z > 0, for all a b c d ∈ Γ 0 (q) satisfies the functional equation and is holomorphic and vanishing at cusps. Then F (z) is called a cusp form of weight k and level q, and has the following Fourier series expansion at infinity Denote the space of all cusp forms of weight k and level q by S k (Γ 0 (q)). For every d|q, the element of the space S k (Γ 0 (d)) can be also considered as an element of the space S k (Γ 0 (q)). The form F ∈ S k (Γ 0 (q)) is called a newform if it is not a cup form of level less than q, and if it is an eigenfunction of all Hecke operators. Then we have that c(1) = 0, therefore, we may assume that F is a normalized newform, i.e., Let s = σ + it be a complex variable. To a newform F , me attach the L-function L(s, F ) defined, for σ > k+1 2 , by Moreover, L(s, F ) has, for σ > k+1 2 , the Euler product over primes is analytically continuable to an entire function and satisfies the functional equation A. Laurinčikas, K. Matsumoto and J. Steuding [1] obtained a limit theorem for the function L(s, F ) and applied it for the investigation of the universality of L(s, F ). Let D = {s ∈ C: k 2 < σ < k+1 2 }, and H(D) denote the space of analytic functions on D equipped ewith the topology of uniform convergence on compacta. Let B(S) stand for the class of Borel sets of the space S. Define where γ p = {s ∈ C: |s| = 1} for all primes p. With the product topology and pointwise multiplication, the torus Ω is a compact topological Abelian group, therefore, on (Ω, B(Ω)) the probability Haar measure m H can be defined. This gives the probability space (Ω, B(Ω)). Denote by ω(p) the projection of ω ∈ Ω to the coordinate space γ p , and define on the probability space Let P L be the distribution of L(s, ω, F ), i.e., Denote by meas{A} the Lebesgue measure of a measurable set A ⊂ R. Then the following statement holds.
Our aim is a joint limit theorem for newforms. For j = 1, . . . , r, let F j be a new form of weight k j and level q j , and L(s, F j ) be the corresponding L-function given, On the probability space (Ω, B(Ω), m H ), define C r -valued random element L(σ, ω, F ) by the formula and σ = (σ 1 , . . . , σ r ), F = (F 1 , . . . , F r ). Denote by P L the distribution of L(σ, ω, F ). Then we have the following theorem.
A generalization of Theorem 2 to the space of analytic functions is also possible. We will give only a sketch of the proof of Theorem 2. Let P denote the set of all prime numbers.

Lemma 1. The probability measure
converges weakly to the Haar measure m H on (Ω, B(Ω)) as T → ∞.
Proof of the lemma is given in [1]. Now let σ 1 > 1 2 be fixed, and, for m, n ∈ N, Then the series for L n (s, F j ) and L n (s, ω, F j ) converge absolutely for σ > kj 2 .

Lemma 2.
Suppose that σ j > kj 2 , j = 1, . . . , r. Then the probability measures both converge weakly to the same probability measure on (C r , B(C r ) as T → ∞.
Define one more probability measurẽ Lemma 4. Suppose that σ j > kj 2 , j = 1, . . . , r. Then the measures P T andP T both converge weakly to the same probability measure P on (C r , B(C r )) as T → ∞.
Then, by Lemma 4, where X n is the random element with the distribution P n , and P n is the limit measure in Lemma 4. After this, it is proved that the family of probability measures {P n : n ∈ N} is tight. Hence, by the Prokhorov theorem, it is relatively compact. Thus, there exists a sequence {P n k } ⊂ {P n } such that P n k converges weakly to a certain probability measure P . In other words, Define X T (σ) = L(σ + iθT, F ).
Proof of Theorem 2. In view of Lemma 4, it suffices to prove that P coincides with P L . For this, elements of the ergodic theory is applied.