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Register a new userArithmetic Progressions of Squares and Multiple Dirichlet Series
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We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.
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Given an abelian group $G$, it is natural to ask whether there exists a permutation $pi$ of $G$ that destroys all nontrivial 3-term arithmetic progressions (APs), in the sense that $pi(b) - pi(a) eq pi(c) - pi(b)$ for every ordered triple $(a,b,c) in G^3$ satisfying $b-a = c-b eq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/Omega_2(G)$ has the same cardinality as $G$, where $Omega_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = mathbb{Z}/nmathbb{Z}$ for all $n eq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(mathbb{Z}/pmathbb{Z})^k$ if and only if $p$ is odd and $(p,k) otin {(3,1),(5,1), (7,1)}$.
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
A multiple Dirichlet series in two variables is constructed as a Mellin transform of a higher order Eisenstein series. It is shown to extend to a meromorphic function and satisfy two independent functional equations.
Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorphic continuation.
There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-Burmann formula for power series. The proof is probabilistic in nature and is based on Kendalls identity, which arises in the fluctuation theory of Levy processes.
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