1,493 research outputs found
An alternative to Riemann-Siegel type formulas
Simple unsmoothed formulas to compute the Riemann zeta function, and
Dirichlet -functions to a power-full modulus, are derived by elementary
means (Taylor expansions and the geometric series). The formulas enable
square-root of the analytic conductor complexity, up to logarithmic loss, and
have an explicit remainder term that is easy to control. The formula for zeta
yields a convexity bound of the same strength as that from the Riemann-Siegel
formula, up to a constant factor. Practical parameter choices are discussed.Comment: 16 page
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
We define the partition and -point functions for a vertex operator algebra
on a genus two Riemann surface formed by sewing two tori together. We obtain
closed formulas for the genus two partition function for the Heisenberg free
bosonic string and for any pair of simple Heisenberg modules. We prove that the
partition function is holomorphic in the sewing parameters on a given suitable
domain and describe its modular properties for the Heisenberg and lattice
vertex operator algebras and a continuous orbifolding of the rank two fermion
vertex operator super algebra. We compute the genus two Heisenberg vector
-point function and show that the Virasoro vector one point function
satisfies a genus two Ward identity for these theories.Comment: 57 Pages, 5 figures. This is an extended version of roughly one half
of arXiv:0712.062
On the Torus Degeneration of the Genus Two Partition Function
We consider the partition function of a general vertex operator algebra
on a genus two Riemann surface formed by sewing together two tori. We consider
the non-trivial degeneration limit where one torus is pinched down to a Riemann
sphere and show that the genus one partition function on the degenerate torus
is recovered up to an explicit universal -independent multiplicative factor
raised to the power of the central charge.Comment: 18 page
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