Selberg-Delange Method And Their Applications

In analytic number theory,we usually estimate the summatory function of certain arithmetic function with Dirichlet series that can be expressed in terms of ?(s).For example,an asymptotic formulas of prime number theory is obtained by Perron for-mulae to the function-?'(s)/?(s)and properties of ?(s).While the Selberg-Delange method is a double extension in analytic method used in the proof of the prime number theory.On the one hand the method can deal with Dirichlet series associated arith-metic function having singularities which are not poles.On the other hand the method can exhibit a kind of stability for the phenomenon,in the sense of an invariance in the nature of the asymptotic formulae estimate for two Dirichlet series whose ratio is a sufficiently regular analytic funciton.Applying this method to some problems arising from number theory,we obtain some results as follows:1?The average distribution of divisors in arithmetic progressions:The limit of distribution function associated the average distribution of divisors in arithmetic pro-gressions is the distribution of the arcsine law under some conditions,and an asymptotic formulas is given,which generalizes the result investigated by Deshouillers,Dress and Tenenbaum.2?The average distribution of divisors in square-free numbers:The limit of distri-bution function associated the average distribution of divisors in square-free numbers is the distribution of the arcsine law,and an asymptotic formula is given,which gen-eralizes the result investigated by Deshouillers,Dress and Tenenbaum.3?Mean value estimates related to some certain multiplicative function in pro-gressions to prime-power modulus:According to the famous result of Gallagher related zero-free region of L function for prime-power modulus,mean value estimate of some certain multiplicative function in progressions to prime-power modulus is given under some conditions,where the modulus is "large" comparatively.4?Distribution of values of Euler's function over arithmetic progressions:According to the Selberg-Delange theorem,distribution of values of Euler's function over arith-metic progressions is given under the conditions of"small" modulus,and an asymptotic formula is obtained,which generalizes the result investigated by Bateman.