Maximal multiplicative properties of partitions

Extending the partition function multiplicatively to a function on partitions, we show that it has a unique maximum at an explicitly given partition for any $n\neq 7$. The basis for this is an inequality for the partition function which seems not to have been noticed before.Comment: 5 pages; in replacement: one typo in References corrected. To appear in: Annals of Combinatoric

Laws of inertia in higher degree binary forms

We consider representations of real forms of even degree as a linear combination of powers of real linear forms, counting the number of positive and negative coefficients. We show that the natural generalization of Sylvester's Law of Inertia holds for binary quartics, but fails for binary sextics.Comment: 13 page

Special values of shifted convolution Dirichlet series

In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution $L$-functions by defining shifted convolution $L$-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.

Higher Width Moonshine

\textit{Weak moonshine} for a finite group $G$ is the phenomenon where an infinite dimensional graded $G$-module $$V_G=\bigoplus_{n\gg-\infty}V_G(n)$$ has the property that its trace functions, known as McKay-Thompson series, are modular functions. Recent work by DeHority, Gonzalez, Vafa, and Van Peski established that weak moonshine holds for every finite group. Since weak moonshine only relies on character tables, which are not isomorphism class invariants, non-isomorphic groups can have the same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width $s\in\mathbb{Z}^+$. For each $1\leq r\leq s$ and each irreducible character $\chi_i$, we employ Frobenius' $r$-character extension $\chi_i^{(r)} \colon G^{(r)}\rightarrow\mathbb{C}$ to define \textit{width $r$ McKay-Thompson series} for $V_G^{(r)}:=V_G\times\cdots\times V_G$ ($r$ copies) for each $r$-tuple in $G^{(r)}:=G\times\cdots\times G$ ($r$ copies). These series are modular functions which then reflect differences between $r$-character values. Furthermore, we establish orthogonality relations for the Frobenius $r$-characters, which dictate the compatibility of the extension of weak moonshine for $V_G$ to width $s$ weak moonshine.Comment: Versions 2 and 3 address comments from the referee

Does Pro-population Policy Raise Per Capita Consumption?

We theoretically analyze the effects of a child allowance, an improvement in the efficiency of child rearing and a labor income tax on the fertility rate and per capita consumption. The effects on per capita consumption are opposite in the absence, and the presence, of unemployment. For example, a child allowance urges people to have more children and allocate more labor to child rearing, decreasing labor supply for the purpose of commodity production. Therefore, under full employment, it decreases per capita consumption. In the presence of unemployment, however, it reduces the deflationary gap and hence stimulates per capita consumption.