| In part I of this paper, we studied the mean value theorems of standard L-functions attached to Ikeda Lift. We proved the mean value theorems of the fourth moment, and the2m-th moment mean value theorem in special cases.In part II of this paper, we studied the zeta functions of graphs, the distribution of poles of Ihara zeta functions, and the Graph Theory Riemann Hypothesis (4.11) and Weak Graph Theory Riemann Hypothesis (4.12). We proved theorems about the distribution of poles of Ihara zeta functions for a class of irregular graphs.Siegel modular form is the first example of multi-variable holomorphic modular forms. This is a very important and active modern research area, which combines the methods of modern number theory, complex analysis and algebraic geometry.Let f be a Hecke-eigen cusp form of weight2κ with respect to SL(2,Z), and n be a positive integer with n≡κ (mod2). Then Ikeda [19] proved that there exists a Siegel cusp form F0of weight κ+n with respect to Sp(2n,Z), which is called Ikeda lift of f, such that the Standard L-function of F0is We studied the mean value theorems of the above formula. Let m be a positive integer andFor m=2,we proved Theorem2.1for the fourth moment in Chapter2. Theorem2.1.Let T>2, l be positiue integer1) For n+1<l<2n,we have I4(σ；T)=T4(l+1-2σ)(2n-l)+1,-n+l+8-1<σ-n+l+2-1, I4(σ；T)=T4(l+1-2σ)(2n-l)+μ(σ+n-l-2-1)+ε,-n+l<σ≤-n+l+8-1, where μ(σ)=11-8σ-16(1-σ)=2-11-8σ-6.2)For n+1≤l≤2n,we have T4(2n-l)2+1log-8T《I4(-n+l+2-1;《T4(2n-l)2+1log8T.3) For n+2≤l≤2n,we have. I4(σ；T)《T4(l+2-2σ)(2n-l)+8(l-n-σ)+μ(σ+n-l+2-1)+ε,-n+l-8-1≤σ<-n+l, I4(σ；T)=T4(l+1-2σ)(2n-1)+8(l-n-σ)+1,-n+l-2-1<σ<-n+l-8-1.4) For n+2≤l≤2n,we have. I4(-n+T)《T4(l+1-2σ)(2n-l)+2-3+ε.5) WE have I4(1；T)《T4n(n-1)+2-3+ε.7) We have T4n2+1(logT)-12《I4(2-1,T)《T4n2+1(logT)20. In Chapter3,by using new methods,we proved Theorem3.1and Thceorem3.2on the lines σ=-n+l+2-1(n≤l≤2n).Theorem3.1. For n+1≤l≤2n,we have I4(-n+l+2-1;T)+T4(2n-l)2+1.Theorem3.2. T4n2+1log4T(loglogT)-16《I4(2-1；T)《T4n2+1log4T(loglogT)16. In Chapter3,we also proved the2m-th mean value theorm:Theorem3.3. For n+1≤l≤2n,we have I2m(-n+l+2-1；T)=T2m(2n-l)2-1.In Chapter4,we studied the zeta functions of graphs,the graph theory analogs of Riemann HYpothesis,and proved twe theorems about the distribu-ti0n of poles of Ihara zeta functions for a class of irregular graphs.zeta functions of graphs first appeared in tho work of Ihara [18] on p-adic groups in the1960s. Serre[34] made the connection with graph theory. We can propose two types of Riermman Hypothesis for Ihara zeta functions in graph theory:Graph Theory Riemman Hypothesis(4.11) and Weak Graph Theory Riemaann Hypothesis(4.12).For a class of irregular graphs,we proved the following theorem:Theorem4.1.Irregular graph G is a simple graph with n vertices. Let q+1be the biggese degree of vertices in graph G,an p+1be the smallest de of vertices in graph G,if grap G satisfies the following two conditions:1) There exists two degree n-1vertices in graph G,2) and p2>q. Then,the Ihara zeta function of graph G does not satisfy the Graph Theory Riemann Hypothesis(4.11).For the irregular graph which is obtained by deleting any two edges from complete graph Kn, we can prove better results: Theorem4.2. Let Kn be the complete graph with n vertices. Let Kn11be the irregular graph which is obtained by deleting two edges from Kn (n≥6). Then Kn11has the following properties:1) It satisfies naive Ramanujan Inequality (4.13), and is Ramanujan in the sense of Lubotzky;2) The Ihara zeta function of Kn11satisfies the Weak Graph Theory Rie-mann Hypothesis (4.12), but contradicts the Graph Theory Riemann Hypothe-sis (4.11). |
