2 research outputs found
Explorations in the theory of partition zeta functions
We introduce and survey results on two families of zeta functions connected
to the multiplicative and additive theories of integer partitions. In the case
of the multiplicative theory, we provide specialization formulas and results on
the analytic continuations of these "partition zeta functions", find unusual
formulas for the Riemann zeta function, prove identities for multiple zeta
values, and see that some of the formulas allow for -adic interpolation. The
second family we study was anticipated by Manin and makes use of modular forms,
functions which are intimately related to integer partitions by universal
polynomial recurrence relations. We survey recent work on these zeta
polynomials, including the proof of their Riemann Hypothesis.Comment: 41 pages, to appear in Exploring the Riemann Zeta Function, 190 years
from Riemann's Birth, Springer, editors: H. Montgomery, A. Nikeghbali, and M.
Rassia
Eulerian series, zeta functions and the arithmetic of partitions
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the
intersection of the additive and multiplicative branches of number theory,
bringing together ideas from partition theory, -series, algebra, modular
forms and analytic number theory. We present a natural multiplicative theory of
integer partitions (which are usually considered in terms of addition), and
explore new classes of partition-theoretic zeta functions and Dirichlet series
-- as well as "Eulerian" -hypergeometric series -- enjoying many interesting
relations. We find a number of theorems of classical number theory and analysis
arise as particular cases of extremely general combinatorial structure laws.
Among our applications, we prove explicit formulas for the coefficients of the
-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical
physics related to quasi-modular forms; we prove partition formulas for
arithmetic densities of certain subsets of the integers, giving -series
formulas to evaluate the Riemann zeta function; we study -hypergeometric
series related to quantum modular forms and the "strange" function of
Kontsevich; and we show how Ramanujan's odd-order mock theta functions (and,
more generally, the universal mock theta function of Gordon-McIntosh)
arise from the reciprocal of the Jacobi triple product via the -bracket
operator, connecting also to unimodal sequences in combinatorics and quantum
modular-like phenomena.Comment: Ph.D. dissertation (2018, Emory University, advisor Ken Ono)
including joint work with Amanda Clemm, Marie Jameson, Ken Ono, Larry Rolen,
Maxwell Schneider and Ian Wagner, 228 page