Download Limit Theorems for the Riemann Zeta-Function by Antanas Laurincikas PDF
By Antanas Laurincikas
The topic of this e-book is probabilistic quantity conception. In a large feel probabilistic quantity concept is a part of the analytic quantity thought, the place the tools and ideas of likelihood idea are used to check the distribution of values of mathematics gadgets. this can be often complex, because it is tough to claim whatever approximately their concrete values. the reason is, the subsequent challenge is generally investigated: given a few set, how frequently do values of an mathematics item get into this set? It seems that this frequency follows strict mathematical legislation. right here we find an analogy with quantum mechanics the place it really is most unlikely to explain the chaotic behaviour of 1 particle, yet that enormous numbers of debris obey statistical legislation. The gadgets of research of this booklet are Dirichlet sequence, and, because the name indicates, the most recognition is dedicated to the Riemann zeta-function. In learning the distribution of values of Dirichlet sequence the susceptible convergence of chance measures on various areas (one of the main asymptotic chance concept equipment) is used. the appliance of this technique was once introduced via H. Bohr within the 3rd decade of this century and it used to be carried out in his works including B. Jessen. extra improvement of this concept used to be made within the papers of B. Jessen and A. Wintner, V. Borchsenius and B.
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Extra resources for Limit Theorems for the Riemann Zeta-Function
Example text
The Besicovitch Spaces For q ~ 1, let Lq be a set of functions f(x) such that lf(x)lq is integrable over [-T, T] for each T > 0. For the functions f(x) E Lq, we put T Mqf = 1/q f~oo ( 2~ JIJ P(f, fo) < Then we can find n1 8}. 5) Let Bc5 n> n2 = {f: p(f, 0) < 8}. 6) Let Qn(A) = IP'(Rn E A), A E B(H(D)). 2) that P = Pn * Qn. 6, we find that P(Au) = J Pn(Au - g)Qn(dg) H(D) ~ Pn(A8) ~ J Pn(Au- g)Qn(dg) B8 j B6 Qn(dg) = Pn(A8)Qn(B8) > 0 ELEMENTS OF THE PROBABILITY THEORY 25 for n ~ n3 = max(nJ,nz). In consequence, foE Sx. 7) Since the random elements X1, ... 7 that SLn is the closure of the set of all f E H(D) which can be written as a sum n f = L fm, fm E Sxon• m=l Now from the definition of LimSLn we deduce that iff E Sx then there exists a sequence {gn} such that Yn E S Ln and limn--+oo Yn = f. 17) proves the theorem. 7. Some Estimates for the Dirichlet Polynomials The Dirichlet polynomials, which we called trigonometric polynomials, have already been studied in the previous sections of the present chapter. In general, the Dirichlet polynomial is a partial sum of some Dirichlet series. 1. The estimate is valid uniformly in (7, 1/2 ~ (7 ~ 1. Proof is the same as in (Titchmarsh, 1951) and uses the estimate which is uniform in (7, 1/2 ~ (7 ~ 1. We will frequently use the following mean value theorem for Dirichlet polynomials.