# Research in Number Theory

Research in Number Theory > 2019 > 5 > 4 > 1-17

*A*isogenous to $$B^r$$ Br , where the characteristic polynomial

*g*of Frobenius of

*B*is an ordinary square-free

*q*-Weil polynomial, for a power

*q*of a prime

*p*, or a square-free

*p*-Weil polynomial with no real roots. Under some extra assumptions on the polynomial

*g*we give an explicit description of all...

Research in Number Theory > 2019 > 5 > 4 > 1-17

Research in Number Theory > 2019 > 5 > 4 > 1-14

*p*of the number of positive integers

*a*such that the fraction

*a*/

*p*can be written as the sum of three unit fractions.

Research in Number Theory > 2019 > 5 > 4 > 1-8

*m*shifted inequality $$\begin{aligned} p(a) \, p(b) \ge p(a+b+m-1) \end{aligned}$$ p(a)p(b)≥p(a+b+m-1) where

*p*(

*n*) is the

*n*th partition number, and $$a,b,m \in \mathbb {N}_0$$ a,b,m∈N0 with

*a*,

*b*positive. The inequality was first studied by Bessenrodt–Ono for $$m=1$$ m=1 . We finally suggest...

Research in Number Theory > 2019 > 5 > 4 > 1-23

*C*/

*k*of $$\ell $$ ℓ -power degree, we describe the field generated by the $$\ell $$ ℓ -power torsion of the Jacobian variety in terms of the branch set and reduction type of

*C*(and hence, in terms of data determined by a suitable affine model of

*C*). If the Jacobian is good away from $$\ell $$ ℓ and the branch set is defined over a...

Research in Number Theory > 2019 > 5 > 4 > 1-8

*L*-functions at $$s=1$$ s=1 of some theta products of weight 3, and express them in terms of special values of generalized hypergeometric functions.

Research in Number Theory > 2019 > 5 > 4 > 1-21

*j*-function—that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of $$q=e^{2\pi i z}$$ q=e2πiz , or an

*I*-Bessel function). We show that this phenomenon does not occur...

Research in Number Theory > 2019 > 5 > 4 > 1-14