The functional equations of Langlands Eisenstein series for SL(n,Z)

In this paper,we present a very simple explicit description of Langlands Eisenstein series for SL(n,Z).The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain divisor sums and certain Whittaker functions that appear in the Fourier coefficients of the Eisenstein series.We conjecture that the functional equations are unique up to a real affine transformation of the s variables defining the Eisenstein series and prove the uniqueness conjecture in certain cases.

Representations by Certain Sextenary Quadratic Forms Whose Coefficients Are 1, 2, 3 and 6

Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.

Modular Invariance and Anomaly Cancellation Formulas in Odd Dimension Ⅱ

By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on(4 r-1)-dimensional manifolds with no assumption that the 3 rd de-Rham cohomology of manifolds vanishes. These anomaly cancellation formulas generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds. We also generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds and the Han–Yu rigidity theorem to the(a, b) case.

On Three Differential Equations Associated with Chebyshev Polynomials of Degrees 3,4 and 6

We shall study the differential equation y^l2=Tn（y）-（1-2μ2）;where μ2 is a constant, Tn（x） are the Chebyshev polynomials with n = 3,4,6. The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on 2F1 （1/4, 3/4; 1; z）, 2F1 （l/3, 2/3; 1; z）, 2F1 （1/6, 5/6; 1; z） and provide a unified approach to a set of identities of Rmanujan involving these hypergeometric functions.

Anomaly cancellation and modularity, II:The E_8× E_8 case

In this paper we show that both of the Green-Schwarz anomaly factorization formula for the gauge group E_8 × E_8 and the Horava-Witten anomaly factorization formula for the gauge group E_8 can be derived through modular forms of weight 14. This answers a question of Schwarz. We also establish generalizations of these factorization formulas and obtain a new Hoˇrava-Witten type factorization formula.

An Inner Product Formula on Two-dimensional Complex Hyperbolic Space

We study the Eisenstein series for a convex cocompact discrete subgroup on a two-dimen- sional complex hyperbolic space Hc^2. We find an inner product formula which gives the connection between Eisenstein series and automorphic Green functions on a two-dimensional complex hyperbolic space HC^2. As an application of our inner product formula, we obtain the functional equations of Eisenstein series.