Complex Valued Multiplicative Functions with Bounded Partial Sums

In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a maximal, well-behaved solution to the multiplicative Poisson equation, defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.

1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.

We show that the partial sums of the power series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so, we obtain information about the zeros of the partial sums. We will only assume that these entire functions have a certain asymptotic behavior at infinity. With this information, we will partially verify for this class of functions a conjecture on the location of the zeros of their partial sums known as the Saff–Varga width conjecture. Numerical results and figures are included to illustrate the results obtained for several well-known functions including the Airy functions and the parabolic cylinder functions.

The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions An=f∈HU:fz=z+an+1zn+1+…,z∈U written simply A when n=1, the newly introduced class involves functions f∈A considered in the study due to their special properties. The aim of this paper is to present the outcomes of the study performed on the new class, which include a coefficient inequality, a distortion theorem and extreme points of the class. The starlikeness and convexity properties of this class are also discussed, and partial sums of functions from the class are evaluated in order to obtain class boundary properties.

Domain of meromorphy of a class of Bateman-like totient zeta functions

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice \(\Lambda \subseteq \mathbb {R}^d\) inside the d-sphere of radius R. In contrast to previous works, we obtain error terms with implied constants depending only on d. Secondly, let \(\phi (s) = \sum _n a(n) n^{-s}\) be a ‘well behaved’ zeta function. A classical method of Landau yields asymptotics for the partial sums \(\sum _{n < X} a(n)\), with power saving error terms. Following an exposition due to Chandrasekharan and Narasimhan, we obtain a version where the implied constants in the error term will depend only on the ‘shape of the functional equation’, implying uniform results for families of zeta functions with the same functional equation.

We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halasz’s method in classical probabilistic number theory.

In Geometric Function Theory, it is well known that the familiar Koebe function f(z) = z/(1 − z)2 is the extremal function for the class 𝒮* of starlike functions in the open unit disk 𝕌 and also that the function g(z) = z/(1 − z) is the extremal function for the class 𝒦 of convex functions in the open unit disk 𝕌. However, the partial sum fn (z) of f(z) is not starlike in 𝕌 and the partial sum gn (z) of g(z) is not convex in 𝕌. The aim of the present paper is to investigate the starlikeness and convexity of these partial sums fn (z) and gn (z). Computational and graphical usages of Mathematica (Version 4.0) as well as geometrical descriptions of the image domains in several illustrative examples are also presented. E-mail: owa@math.kindai.ac.jp

In this paper, we define a class of meromorphic p-valent functions and study some properties as coefficient inequality, closure theorem , growth and distortion bounds , arithmetic mean, radius of convexity, Convex linear combination and partial sums .

Let be the system of polynomials orthonormal on with weight where , , on and ( is the modulus of continuity in ). Consider the class of functions , where Let denote the partial sums of the Fourier series of a function with respect to the system .In the paper, conditions are obtained on the exponents of the functions and and the exponent that are necessary and sufficient for the boundedness in of each of the operators and . Sufficient conditions for the convergence of the partial sums to in the mean and almost everywhere in are revealed as a consequence. It is proved that these conditions are best possible on the class (for in the case of convergence almost everywhere). Estimates of the polynomials and necessary and sufficient conditions for their boundedness in the mean are also obtained.Bibliography: 26 items.

Let ?(x) denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be ?*(x) = -?(x) + 2?(2x)-1/2?(4x). We show that ?T+H,T ?*(t/2?)|?(1/2+it)|2dt&lt;&lt; HT1/6log7/2 T (T2/3+? ? H = H(T) ? T), ?T,0 ?(t)|?(1/2+it)|2dt &lt;&lt; T9/8(log T)5/2, and obtain asymptotic formulae for ?T,0 (?*(t/2?))2|?( 1/2+it)|2 dt, ?T0 (?*(t/2?))3|?(1/+it)|2 dt. The importance of the ?*-function comes from the fact that it is the analogue of E(T), the error term in the mean square formula for |?(1/2+it)|2. We also show, if E*(T) = E(T)-2??*(T/(2?)), ?T0 E*(t)Ej(t)|?(1/2+it)|2 dt &lt;&lt; j,? T7/6+j/4+? (j=1,2,3).

In the present paper, we introduce and explore certain new classes of meromorphic functions related with closed-to-convexity and q-calculus. Such results as coefficient estimates, grow the property and partial sums are derived. It is important to mentioned that our results are generalization of number of existing results in literature.

On the minimax approximation in the class of the univariate piecewise constant functions

Network-based function prediction and interactomics: The case for metabolic enzymes

Multi-label classification and features investigation of antimicrobial peptides with various functional classes