Related papers: On spt-crank type functions

In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms.…

For $\frac12<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight $\omega$, we characterize those inner functions $\Theta$ for which $$\|\Theta'\|_{A^{p,q}_\omega}^q=\int_0^1 \left(\int_0^{2\pi} |\Theta'(re^{i\theta})|^p…

Volterra functions were introduced at the beginning of the twentieth century as solutions of some integral equations of convolution type with logarithmic kernel. Since then, few authors have studied this family of functions and faced with…

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

We use the definition of a star (or Moyal or twisted) product to give a phasespace definition of the $\zeta$-function. This allows us to derive new closed expressions for the coefficients of the heat kernel in an asymptotic expansion for…

We compute effective bounds for $\alpha(n)$, the Fourier coefficients of Ramunujan's mock theta function $f(q)$ utilizing a finite algebraic formula due to Bruinier and Schwagenscheidt. We then use these bounds to prove two conjectures of…

We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoglu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions.…

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…

Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining…

Recently, Nath and Das investigated congruence properties for the second order mock theta function $B(q)$. In their paper, they asked for analytic proofs of three identities on the second order mock theta functions $A(q)$, $B(q)$ and…

The notion of ordinal concavity of utility functions has recently been considered by Hafalir, Kojima, Yenmez, and Yokote in economics while there exist earlier related works in discrete optimization and operations research. In the present…

We denote functions mapping n to the Fourier coefficient of q^n in the expansion of a cusp form as Ramanujan functions. We empirically study the eigenvalues of determinants that represent values of these Ramanujan functions. In some cases,…

We provide a new and very short proof of the fact that a spherical functor between certain triangulated categories induces an autoequivalence.

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar…

With a weighting scheme proportional to t, a traditional stochastic gradient descent (SGD) algorithm achieves a high probability convergence rate of O({\kappa}/T) for strongly convex functions, instead of O({\kappa} ln(T)/T). We also prove…

We define, answering a question of Sarnak in his letter to Bombieri, a symplectic pairing on the spectral interpretation (due to Connes and Meyer) of the zeroes of Riemann's zeta function. This pairing gives a purely spectral formulation of…

In this paper we construct a cyclically invariant Boolean function whose sensitivity is $\Theta(n^{1/3})$. This result answers two previously published questions. Tur\'an (1984) asked if any Boolean function, invariant under some transitive…

In this paper, we study shifted Schur functions $S_\mu^\star$, as well as a new family of shifted symmetric functions $\mathfrak{K}_\mu$ linked to Kostka numbers. We prove that both are polynomials in multi-rectangular coordinates, with…

On page 206 in his lost notebook, Ramanujan recorded an incomplete septic theta function identity. Motivated by the completion of this identity by the second author, we offer cubic and quintic analogues. Using the theory generated by these…

In this paper we set up the theory of acid zeta function and ajoint acid zeta function, based on the theory, we point out a reason to doubt the truth of the Riemann hypothesis and also as a consequence, we give out some new RH equivalences.