Partition Function And Additive Representation Function Under Restricted Conditions

In this thesis we investigate the root partition functions,the colored partition functions and the additive representation functions.The main results are as follows:1.Asymptotic formulas for the root partition functionsLet p(n)denote the number of partitions of n,which is the classical partition function.In 1918,Hardy and Ramanujan gave the asymptotic formula for p(n),that p(n)?1/4n(?)3exp(?(?)2n/3).The partition function p(n)has a long history and it is extended into many partition functions with the restricted condition.Let p(n,k)be the number of partitions of n into exactly k parts.In 1941,Erd6s and Lehner gave its asymptotic formula p(n,k)?(n-1 k-1)/k! when k=o(n1/3).Let A={a1,a2,…,ak} be a finite set of positive integers,and(a1,a2,…,ak)= 1.Let p(n,A)be the number of partitions of n with the parts in A.In 2000,Nathanson gave its asymptotic formula.In this paper,we study the asymptotic formula for the root partition functions.For any positive real number r,let pr(n)be the number of solutions of the equa-tion n-[r(?)a1+[r(?)a1]+…+[r(?)ak]with integers 1 ? a1 ? a2 ? … ? ak.We call pr(n)r-th root partition function.In 2015,we give the upper bound and the lower bound of p2(n),that is,there are two positive constants ?1 and ?1 such that exp(?1n2/3)? p2(n)? exp(?2n2/3).In 2016,Luca and Ralaivaosaona further gave an asymptotic formula for p2(n).In 2016,for any real number r>1,we give the upper bound and the lower bound of pr(n).In the first chapter,for any real number r>1,we give an asymptotic formula for pr(n),that is,where l is the integer with l<r?l +1 and c1,c2,...,cl are computable constants depending only on r.In particular,c1 =(1 + 1/r)(r?(r + 1)?(r + 1))1/r+1.This result has been published in by J.Number Theory.2.Asymptotic formulas for the colored partition functionsMany mathematicians have paid much attention to properties on some special colored partition functions.For example,Chan and Kim et al.considered the 2-colored partition qk(n)whch is the number of 2-color partitions of n where one of the colors appears only in parts that are multiples of k.Chan and Cooper et al.considered a special partition function c(n)which is the number of 4-colored partitions of n with two of the colors appearing only in multiplies of 3.They gave a series of congruences.In this paper,we study the asymptotic formulas for general colored partition functions.Given positive integers 1 = s1<s2<…<sk and l1,l2,...,lk.In the partition of n n = a1 + a2 + … +am,coloring aj(1?j?m)with l1+l2+…+lk colors,which is denoted by 1,2,...,l1+l2…+lk,and the colors l1+l2+…+li+1,l1+l2f+…+li appearing only in multiplies of si(1?i?k),we call the number of this partition(s,l)-colored partition function g(s,l,n),where s=(s1,s2,...,sk),l=(l1,l2,...,lk).In the second chapter,we obtain an asymptotic formula for g(s,l,n)with an explicit error term,that is,for any given ?>0,we have where3.The additive representation functionsFor any set A(?)N and n ? N,let RA(n)be the number of solutions of the equation n = a + b,a,b ? A.The classical Erdos-Turan conjecture says that if RA(n)? 1 for all sufficiently large integers n,then RA(n)is unbounded.But,this conjecture is still open.Many mathematicians have paid much attention to the additive representation function in abelian group.For any finite abelian group G with |G| = m,A(?)G and g E G,let RA(g)be the number of solutions of the equation g = a + b,a,b ? A.Recently,Sandor and Yang proved that,if m ? 36 and RA(n)?1 for all n ?Zm,then there exists n0 ?Zm.such that RA(n0)? 6.In the third chapter,for any finite abelian group G with |G| = m and A C G,we prove that(a)if the number of g ? G with RA(g)= 0 does not exceed 7/32m-1/2(?)10m-1,then there exists g E G such that RA(g)? 6;(b)if 1 ? RA(g)? 6 for all g ? G,then the number of g ? G with RA(g)= 6 is more than 7/32m-1/2(?)m-1.This result has been published in Bull.Aust.Math.Soc.