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Register a new userThe kernel of newform Dedekind sums
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Newform Dedekind sums are a class of crossed homomorphisms that arise from newform Eisenstein series. We initiate a study of the kernel of these newform Dedekind sums. Our results can be loosely described as showing that these kernels are neither too big nor too small. We conclude with an observation about the Galois action on Dedekind sums that allows for significant computational efficiency in the numerical calculation of Dedekind sums.
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In this paper we consider Dedekind type DC sums and prove receprocity laws related to DC sums.
Let $eta$ be the weight $1/2$ Dedekind function. A unification and generalization of the integrals $int_0^infty f(x)eta^n(ix)dx$, $n=1,3$, of Glasser cite{glasser2009} is presented. Simple integral inequalities as well as some $n=2$, $4$, $6$, $8$, $9$, and $14$ examples are also given. A prominent result is that $$int_0^infty eta^6 (ix)dx= int_0^infty xeta^6 (ix)dx ={1 over {8pi}}left({{Gamma(1/4)} over {Gamma(3/4)}}right)^2,$$ where $Gamma$ is the Gamma function. The integral $int_0^1 x^{-1} ln x ~eta(ix)dx$ is evaluated in terms of a reducible difference of pairs of the first Stieltjes constant $gamma_1(a)$.
Let $R$ be an infinite Dedekind domain with at most finitely many units, and let $K$ denote its field of fractions. We prove the following statement. If $L/K$ is a finite Galois extension of fields and $mathcal{O}$ is the integral closure of $R$ in $L$, then $mathcal{O}$ contains infinitely many prime ideals. In particular, if $mathcal{O}$ is further a unique factorization domain, then $mathcal{O}$ contains infinitely many non-associate prime elements.
We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $A$ we prove that [ frac{|(k+1)wedge A|}{|kwedge A|}, le , frac{|A|-k}{k+1} ] whenever $|A|ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.
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