On The Number Of Distinct Prime Factor And The Distribution Of The Largest Prime Factor Of The Partition Function

A partition of a positive integer n is any non-increasing sequence of natural number whose sum is n,the total number of partitions of n is called partition function and denoted by p(n).There are many unsolved conjectures and problems with the partition function,which is closely related to number theory,combinatorics,representation theory and mathematical physics.In this thesis,we study the distinct prime factors and the largest prime factor of the partition function p(n).Let ω(n)denotes the number of different prime factors of n.In 1987,Schinzel and Wirsing proved (?) Our first result is a generalization of the theorem of Schinzel and Wirsing.Suppose R is a positive integer,{br}r=1R is a sequence of non-negative integers satisfying the appropriate conditions.Define F2={0,1},e=(ε1,ε2,…,εR),B=(b1,b2,…,bR).Ifεi ∈ F2,i=1,2,…,R.then (?) where c>0 is a constant.Let P(n)be the largest prime factor of the positive integer n with the convention that P(1)=1.Cilleruelo and Luca proved that the asymptoic density of n satisfying P(p(n))≥log log n is 1.Our second result is an improvent and extension of the theorem of Cilleruelo and Luca.Suppose arithmetic F(n)is a function satisfying (?) where c1,…,c7,a,b,c,d,s are positive constants.We prove that the asymptoic density of n satisfying(?) is 1,where log log n=log2 n,log log log n=log3 n,log log log log n=log4 n.Our method is an estimate on lower bounds for nonzero linear forms in logarithms of algebraic numbers in transcendental number theory.In particular,when F(n)=p(n),we improved the result of Cilleruelo and Luca.The same results hold for F(n)=α(n),where α(n)is the coefficients of the third-order mock theta function of Ramanujan,and for F(n)=c(n),where c(n)is the coefficients of j-invariant function.