Distribution of additive functions and mean values of multiplicative functions

2015 ◽  
pp. 475-510
Keyword(s):  
Multiplicative Functions ◽  
Additive Functions ◽  
Mean Values

Integral Limit Laws for Additive Functions

1973 ◽  
Vol 25 (1) ◽  
pp. 194-203
Author(s):  
J. Galambos
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Mean Value ◽  
Characteristic Functions ◽  
Asymptotic Formulas ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Additive Functions ◽  
Limit Laws ◽  
Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

Mean values of some multiplicative functions

2015 ◽  
Vol 97 (1-2) ◽  
pp. 111-123
Author(s):  
A. A. Sedunova
Keyword(s):  
Multiplicative Functions ◽  
Mean Values

scholarly journals On the asymptotic formulae for some multiplicative functions in short intervals

2015 ◽  
Vol 11 (05) ◽  
pp. 1571-1587 ◽  
Author(s):  
Alisa Sedunova
Keyword(s):  
Short Interval ◽  
Multiplicative Functions ◽  
Density Estimates ◽  
Mean Values ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Free Region ◽  
Complex Integration ◽  
The Mean ◽  
Riemann Ζ Function

We are going to study the mean values of some multiplicative functions connected with the divisor function in short interval of summation. The asymptotics for such mean values will be proved. Considering instead of well-known multiplicative functions, their inverses lead to very weak results of application of standard methods of complex integration. In order to get better estimations, we propose another method which uses as its main tools the density estimates and zero-free region for Riemann ζ-function and Dirichlet L-functions.

Mappings on Decomposable Combinatorial Structures: Analytic Approach

2002 ◽  
Vol 11 (1) ◽  
pp. 61-78 ◽  
Author(s):  
E. MANSTAVIČIUS
Keyword(s):  
Number Theory ◽  
Generating Functions ◽  
Multiplicative Functions ◽  
Combinatorial Structures ◽  
Mean Values ◽  
Taylor Coefficients ◽  
Probabilistic Number ◽  
Generating Series ◽  
Euler Products ◽  
Complex Valued

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

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