# Research in Number Theory

Research in Number Theory > 2015 > 1 > 1 > 1-9

*P*

_{ λ }(

*x*

_{1},

*x*

_{2},…;

*q*) to arrive at expressions of the form ∑ λ : λ 1 ≤ m q a | λ | P 2 λ ( 1 , q , q 2 , … ; q n ) = “Infinite product modular function” $$\sum\limits_{\lambda : \lambda_{1} \leq m} q^{a|\lambda|}P_{2\lambda}(1,q,q^{2},\ldots...

Research in Number Theory > 2015 > 1 > 1 > 1-11

Research in Number Theory > 2015 > 1 > 1 > 1-13

Research in Number Theory > 2015 > 1 > 1 > 1-42

*reified*, i.e., whose value groups have been forced to contain the real numbers. This yields

*reified adic spectra*which provide a framework for an analogue of Huber’s theory compatible with Berkovich’s construction of nonarchimedean...

Research in Number Theory > 2015 > 1 > 1 > 1-13

*E*and a prime

*p*of (good) supersingular reduction, we formulate

*p*-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions

*L*

_{ ♯ }(

*E*,

*T*) and

*L*

_{ ♭ }(

*E*,

*T*). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient...

Research in Number Theory > 2015 > 1 > 1 > 1-18

*R*(

*x*) diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan...

Research in Number Theory > 2015 > 1 > 1 > 1-19

*G*. For non-compact

*G*, these laws generalize the following observations: (1) the normalized Haar measure of the Lie group ℝ + is

*d*

*x*/

*x*and (2) the scale invariance of

*d*

*x*/

*x*implies the distribution of the digits follow Benford’s law. Viewing this scale invariance as left invariance of Haar measure,...

Research in Number Theory > 2015 > 1 > 1 > 1-18

Research in Number Theory > 2015 > 1 > 1 > 1-31

*Mock modular forms*, which give the theoretical framework for Ramanujan’s enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E / Q . We show that mock modular forms which arise from Weierstrass

*ζ*-functions encode the central

*L*-values and

*L*-derivatives which occur in the Birch and Swinnerton-Dyer...

Research in Number Theory > 2015 > 1 > 1 > 1-16

*p*(

*n*). We obtain a formula for the coefficients of the mock modular forms of weight 5/2 in terms of regularized inner products of weakly holomorphic modular forms of weight −1/2,...

Research in Number Theory > 2015 > 1 > 1 > 1-5

Research in Number Theory > 2015 > 1 > 1 > 1-8

Research in Number Theory > 2015 > 1 > 1 > 1-34

Research in Number Theory > 2015 > 1 > 1 > 1-25

*ω*(

*q*) (resp.

*ν*(−

*q*)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function

*ϕ*(

*q*). Congruences for the smallest...

Research in Number Theory > 2015 > 1 > 1 > 1-20

*ζ*

_{ F }(

*s*,

*A*) of a wide ideal class

*A*of a totally real number field

*F*of degree

*n*. This formula relates the constant term in the Laurent expansion of

*ζ*

_{ F }(

*s*,

*A*) at

*s*=1 to a toric integral of a SL n ( ℤ ) ${SL}_{n}({\mathbb {Z}})$ -invariant function log

*G*(

*Z*) along a Heegner cycle in the symmetric space of GL n ( ℝ...

Research in Number Theory > 2015 > 1 > 1 > 1-8

*g*Siegel modular forms modulo a prime

*p*, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases

*g*=1,2, which do not extend to the...

Research in Number Theory > 2015 > 1 > 1 > 1-37

Research in Number Theory > 2015 > 1 > 1 > 1-15

Research in Number Theory > 2015 > 1 > 1 > 1-18

*q*-series proofs of recent results of Imamoğlu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions. Additionally, we discuss an application of this identity to other mock theta functions. Mathematics Subject Classification Primary: 33D15; Secondary: 11F30

Research in Number Theory > 2015 > 1 > 1 > 1-14

*q*-hypergeometric series and mock and partial...