8,837 research outputs found
Asymptotic formulae for partition ranks
Using an extension of Wright's version of the circle method, we obtain
asymptotic formulae for partition ranks similar to formulae for partition
cranks which where conjectured by F. Dyson and recently proved by the first
author and K. Bringmann
Lacunary recurrences for Eisenstein series
Using results from the theory of modular forms, we reprove and extend a
result of Romik about lacunary recurrence relations for Eisenstein series.Comment: 6 pages, more detailed proofs in v3, accepted for publication in
Research in Number Theor
On class invariants for non-holomorphic modular functions and a question of Bruinier and Ono
Recently, Bruinier and Ono found an algebraic formula for the partition
function in terms of traces of singular moduli of a certain non-holomorphic
modular function. In this paper we prove that the rational polynomial having
these singuar moduli as zeros is (essentially) irreducible, settling a question
of Bruinier and Ono. The proof uses careful analytic estimates together with
some related work of Dewar and Murty, as well as extensive numerical
calculations of Sutherland
Special values of shifted convolution Dirichlet series
In a recent important paper, Hoffstein and Hulse generalized the notion of
Rankin-Selberg convolution -functions by defining shifted convolution
-functions. We investigate symmetrized versions of their functions. Under
certain mild conditions, we prove that the generating functions of certain
special values are linear combinations of weakly holomorphic quasimodular forms
and "mixed mock modular" forms.Comment: 18 pages, corrected slight error in main theorem and made according
minor edits in Sections 3.4 and 3.
Random Walks in Rindler Spacetime and String Theory at the Tip of the Cigar
In this paper, we discuss Rindler space string thermodynamics from a thermal
scalar point of view as an explicit example of the results obtained in JHEP
1402 (2014) 127. We discuss the critical behavior of the string gas and
interpret this as a random walk near the black hole horizon. Combining field
theory arguments with the random walk path integral picture, we realize (at
genus one) the picture put forward by Susskind of a long string surrounding
black hole horizons. We find that thermodynamics is dominated by a long string
living at string-scale distance from the horizon whose redshifted temperature
is the Rindler or Hawking temperature. We provide further evidence of the
recent proposal for string theory at the tip of the cigar by comparing with the
flat space orbifold approach to Rindler thermodynamics. We discuss all types of
closed strings (bosonic, type II and heterotic strings).Comment: 54 pages, v2: version accepted for publication in JHE
Near-Hagedorn Thermodynamics and Random Walks: a General Formalism in Curved Backgrounds
In this paper we discuss near-Hagedorn string thermodynamics starting from
the explicit path integral derivation recently found by JHEP 0607 (2006) 031.
Their result is extended and the validity is checked by comparing with some
known exact results. We compare this approach with the first-quantized one-loop
result from the low energy effective field theory and establish correction
terms to the above result.Comment: 38 pages, v2: version accepted for publication in JHE
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