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Showing 1 to 20 of 20 for “"Eisenstein series"”.
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Eisenstein series on the boundary of the disk,
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1970.
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Eisenstein Series, Analogues of the Rogers -Ramanujan Functions, and Partition Identities
Chapter 4 is devoted to finding new partition identities inspired by the work of O. Kolberg and S. Ramanujan. We use functions studied by N. J. Fine and R. J. Evans to construct analogues of modular equations, and then derive new identities satisfied by n=0infinity p(ln + k) qn, where p(n) is the …
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On Colin de Verdière's Proof of the Meromorphic Continuation of Eisenstein Series
… proof of the meromorphic continuation of Eisenstein series. We consider the importance of this result as it relates to ideas relating to the Riemann hypothesis and offer a timeline of other results that influenced this eventual proof.
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Analytic Continuation and Functional Equations of Cuspidal Eisenstein Series for Maximal Cuspidal Subgroups
… of Selberg to the analytic continuation of Eisenstein series by means of Fredholm theory. We set up the machinery for very general algebraic groups and arithmetic subgroups, and obtain the analytic continuation and functional equations of cuspidal Eisenstein series for rank one cuspidal …
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L-functions and J-spectra
The relation between Eisenstein series and the J-homomorphism is an important topic in chromatic homotopy theory at height 1. Both sides are related to the special values of the Riemann ζ-function. Number theorists have studied the twistings of the Riemann ζ-functions and Eisenstein series by …
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ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|
… in the Laurent expansion of a non-holomorphic Eisenstein series at one of its poles. Asai generalised the limit formula to Eisenstein series of level one defined for a number field with class number one and obtained a function analogous to the logarithm of the absolute value of the eta …
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Topics on the Spectral Theory of Automorphic Forms
We study the analytic properties of the Eisenstein Series of $frac {1}{2}$-integral weight associated with the Hecke congruence subgroup $Gamma_0(4)$. Using these properties we obtain asymptotics for sums of certain Dirichlet $L$-series. We also obtain a formula reducing the study of Selberg's …
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Higher Siegel-Weil formulae over function fields
… of these cycles to higher derivatives of Siegel-Eisenstein series. In this thesis, we generalize their result in two directions: we 1) prove a higher Siegel-Weil formula for unitary groups for corank-1 degenerate coefficients, and 2) introduce analogous cycles on moduli spaces of symplectic …
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Bounds for the Spectral Mean Value of Central Values of L-functions
… L-functions at s = 1/2, which is an analogue for Eisenstein series of X. Li's result for Hecke-Maass forms.
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Ramanujan's formulas for the coefficients in the power series expansions of certain modular forms
… formulas for the coefficients in the power series expansions of certain modular forms. We prove his formulas for the coefficients of 1/$E\sb4, E\sb4/E\sb6$ and other functions involving the Eisenstein series $E\sb4, E\sb6$ and $E\sbsp{2}{*}$. These formulas are stated, without proof, in a …
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Boundary Value Problems Of Spaces Of Automorphic Forms
… appearing as a compact period of cuspidal-data Eisenstein series on GL 4 . These ideas have their origins in re- sults of Hejhal and Colin de Verdi ́ere. In parallel with the GL(2) case, the corresponding pair-correlation and triple-correlation results limit the fraction of on-the-line zeros …
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An Arithmetic-geometric Reciprocity between Theta Functions Attached to Real and Imaginary Quadratic Fields
… theta functions are special values of classical Eisenstein series at CM points.
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Co-rank 1 Arithmetic Siegel--Weil
… in near-central first derivatives of unitary Eisenstein series Fourier coefficients. The key input is a new limiting method at all places. On the analytic side, the limit relates local Whittaker functions on different groups. On the geometric side at nonsplit non-Archimedean places, the limit …
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Congruences in modular, Jacobi, Siegel, and mock modular forms with applications
… in the Fourier coefficients of certain ratios of Eisenstein series. We also determine the exact number of holomorphic modular forms with Ramanujan congruences when the weight is large enough. In a chapter based on joint work with Olav Richter, we study Ramanujan congruences in the coefficients of …
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Identities between Hecke Eigenforms
… that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. Assuming Maeda's conjecture, we prove that the Petersson inner product $\langle f^2,g\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\mathhbb{Z})$ of weight $k$ and $2k$, …
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Analogues of Dedekind Sums
… Finally, with a generalization of an Eisenstein series, we develop transformation formulas involving new sums that are generalizations of the Dedekind sum. We prove a reciprocity law for one of the new sums.
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L-functions in Number Theory
… newforms is twisted by the derivative of some Eisenstein series. Finally, we consider the Artin L-functions attached to irreducible 4-dimensional S5-Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an …
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Higher Dimensional Adeles, Mean-Periodicity and Zeta Functions of Arithmetic Surfaces
… “boundary function” plays the role of an Eisenstein series. 2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective …
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Conformally Invariant Operators in Higher Spin Spaces
… manifolds. This can be done with the help of Eisenstein series as in [31].</p>
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Transformation formulas associated with integrals involving the Riemann Ξ-function and rank-crank type PDEs in partition theory
… through certain bilateral basic hypergeometric series identities due to M.~Jackson and S.H.~Chan. The motivation comes from the fact that the original Rank-Crank PDE due to A.O.L.~Atkin and F.G.~Garvan was obtained through an identity of Atkin and H.P.F.~Swinnerton-Dyer, which is a special case …