649 research outputs found
Thirty-two Goldbach Variations
We give thirty-two diverse proofs of a small mathematical gem--the
fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also
discuss various generalizations for multiple harmonic (Euler) sums and some of
their many connections, thereby illustrating both the wide variety of
techniques fruitfully used to study such sums and the attraction of their
study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory
material added and material on inequalities, Hilbert matrix and Witten zeta
functions. Errors in the second section on Complex Line Integrals are
corrected. To appear in International Journal of Number Theory. Title change
Gravity, strings, modular and quasimodular forms
Modular and quasimodular forms have played an important role in gravity and
string theory. Eisenstein series have appeared systematically in the
determination of spectrums and partition functions, in the description of
non-perturbative effects, in higher-order corrections of scalar-field spaces,
... The latter often appear as gravitational instantons i.e. as special
solutions of Einstein's equations. In the present lecture notes we present a
class of such solutions in four dimensions, obtained by requiring (conformal)
self-duality and Bianchi IX homogeneity. In this case, a vast range of
configurations exist, which exhibit interesting modular properties. Examples of
other Einstein spaces, without Bianchi IX symmetry, but with similar features
are also given. Finally we discuss the emergence and the role of Eisenstein
series in the framework of field and string theory perturbative expansions, and
motivate the need for unravelling novel modular structures.Comment: 45 pages. To appear in the proceedings of the Besse Summer School on
Quasimodular Forms - 201
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