The Mean Value Problem Of A Class Of Arithmetic Functions On A Short Interva

Divisor function and von Mangoldt function are important arithmetic functions in analytic number theory.In part of this thesis,we study the distribution of divisor factors of integers that represent the sum of mixed powers of four variables.Due to the irregular distribution of divisor factors,we consider the mean value of divisor function of the sum of mixed powers of four variables,and obtain the corresponding mean value estimation.In the other part of this thesis,we study the problem of prime numbers that represent the sum of mixed powers of three integers by von mangoldt function.By means of the Bombieri type theorem of the exponential sum of prime variables in short intervals,we obtain the method of the enlarged major arcs,and establish the asymptotic formula of the corresponding mean value problem by combining the circle method and the trigonometric sum estimation.This result is a further exploration of prime distribution and enriches the research results in this field.The first chapter mainly introduces the research background and main conclusions.The second chapter briefly illustrate the circle method and its application by the proof of the three prime theorem,and proves the main lemmas.In the next chapter,through the circle method and the exponential sum method,according to the different power,we give the proof of the theorem of the mean value of divisor function of mixed power of four variables in short intervals.In the last chapter,by using the circle method and the method of enlarged major arcs,we get the theorem of the mean value of von Mangoldt function of mixed power of three variables in short intervals.