The paper presents universality of some analytic functions, which is a very exceptional and useful property of zeta and L-functions. The property for the Riemann zeta-function was discovered by S. M. Voronin. Later, many mathematicians, such as S. M. Gonek, A. Reich, B. Bagchi, A. Laurinčikas, K. Matsumoto, R. Garunkštis, J. Steuding and others improved and generalized Voronin’s theorem. Physicists examine numerous models from different branches of physics (from classical mechanics to statistical physics) where this function plays an integral role.
In the paper, a short survey on universality results for L-functions of elliptic curves over the field of rational numbers is given and weighted universality theorem is proven. All stated universality theorems are of continuous type. The proof of the universality for L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.
Straipsnyje įrodyta elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo teorema silpnojo tikimybini ų matų konvergavimo prasme analizinių funkcijų erdvėje. Nagrinėjamas analizinės funkcijos aproksimavimas postūmiais L‘E (s + imh), čia kompleksinio kintamojo menamosios dalies postūmiai įgyja reikšmes iš diskrečiosios aibės, pavyzdžiui, aritmetinės progresijos. Fiksuotas skaičius h > 0 pasirenkamas taip, kad exp{2πk/h} būtų iracionalusis skaičius visiems k ∈ Z \{0} . Elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo įrodymas remiasi diskrečiąja ribine teorema tikimybinių matų silpnojo konvergavimo prasme analizinių funkcijų erdvėje.
In the paper, we prove the discrete limit theorem in the sense of the weak convergence of probability measures in the space of analytic on DV = {s ∈ C : 1 < σ < 3/2, |t| < V} functions for L-functions of elliptic curves LE(s). The main statement of the paper is as follows. Let h > 0 be a fixed real number such that exp {2πk/h} is an irrational number for all k∈Z\{0}. Then the probability measure μN(LE(s + imh)∈A), A ∈ B(H(DV)), converges weakly to the measure PLE as N→∞.
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