Asymptotics for the Enumeration of Commuting Matrices over Finite Fields
We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.
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arXiv subjects
Johann Franke
Publications and source records attributed to Johann Franke.
L-series of Eisenstein series vanishing at critical values
Using the relations between rational functions and Eisenstein series, as well as the inferences for cotangent sums and period polynomials, we work out a precise description for Eisenstein series whose $L$-series vanish at certain critical values. This is possible for small weights compared to the level of the Eisenstein series. For large weights we give a partial result and determine subspaces with simultaneous vanishing properties.
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On the Proportion of Coprime Fractions in Number Fields
In this paper we determine the asymptotic density of coprime fractions in those of the reduced fractions of number fields. When ordered by norms of denominators, we count a fraction as soon as it ``appears'' for the first time and no later. The natural density of coprime fractions in the set of reduced fractions may then be computed using well-known facts about Hecke $L$-functions. Furthermore, we draw some connections to the modular group and Heegner points.
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Asymptotics of commuting $\ell$-tuples in symmetric groups and log-concavity
Denote by $N_{\ell}(n)$ the number of $\ell$-tuples of elements in the symmetric group $S_n$ with commuting components, normalized by the order of $S_n$. In this paper, we prove asymptotic formulas for $N_\ell(n)$. In addition, general criteria for log-concavity are shown, which can be applied to $N_\ell(n)$ among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form $c(a)c(b) > c(a+b)$ for certain families of sequences $c(n)$.
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On the number of irreducible representations of $\mathfrak{su}(3)$
In this note, we use a variant of the hyperbola method to prove an asymptotic expansion for the summatory function of the number of irreducible $\mathfrak{su}(3)$-representations of dimension $n$. This is a natural companion result to work of Romik, who proved an asymptotic formula for the number of unrestricted $\mathfrak{su}(3)$-representations of dimension $n$.
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Asymptotic expansions for partitions generated by infinite products
Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $\mathcal{L}$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree $n$ of $\mathfrak{so}{(5)}$. We also study the Witten zeta function $ζ_{\mathfrak{so}{(5)}}$, which is of independent interest.
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Sign changes in statistics for plane partitions
Recent work of Cesana, Craig and the third author shows that the trace of plane partitions is asymptotically equidistributed in residue classes mod $b$. Applying a technique of the first two authors and Garnowski, we prove asymptotic formulas for the secondary terms in this equidistribution, which are controlled by certain complex numbers generated by a twisted MacMahon-type product. We further carry out a similar analysis for a statistic related to plane overpartitions.
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Asymptotics for the twisted eta-product and applications to sign changes in partitions
We prove asymptotic formulas for the complex coefficients of $(ζq;q)_\infty^{-1}$, where $ζ$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions of $n$ with largest part congruent $a$ modulo $b$. Our results imply that, as $n \to \infty$, the difference $p(a_1,b,n)-p(a_2,b,n)$ for $a_1 \neq a_2$ oscillates like a cosine, when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of $\{p(a,b,n)\}_{1 \leq a \leq b}$.
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An asymptotic formula for the number of $n$-dimensional representations of $\mathrm{SU}(3)$
We prove an asymptotic formula for the number of $n$-dimensional representations of the group $\mathrm{SU}(3)$. Main tools for the proof are Wright's Circle Method and the Saddle Point Method.
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