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Keywords:
Dedekind sum; Hurwitz zeta-function; Lerch zeta-function; vector space structure; generalized inner product
Dedekind sum; Hurwitz zeta-function; Lerch zeta-function; vector space structure; generalized inner product
Summary:
As a far generalization of the Dedekind sum with the product of periodic Bernoulli polynomials, Mikolás introduced the Dedekind type sum $\mathcal {M}_c^{a,b}(w,z)$ with the product of the Hurwitz zeta-functions $\zeta (s,x)$, $0<x\le 1$. We adopt the motivation suggested by Mikolás that the Dedekind sum is a generalized inner product in the second variable. The Hurwitz zeta-function has a simple pole at $s=1$ and cannot assume the value $x=0$ while its counterpart, the Lerch zeta-function $\ell _s(x)=\ell (s,x)$, is more tractable and we study the Dedekind type sum $\mathcal {L}_c^{a,b}(w,z)$ with the product of the Lerch zeta-functions. We establish a striking identity between these Dedekind type sums to the effect that $\mathcal {M}_c^{a,b}(w,z)$ with a correction term is a constant multiple of $\mathcal {L}_c^{a,b}(w,z)$ -- the base change formula. This implies a new expression for the ordinary Dedekind sum in terms of the one with Apostol's generalized Bernoulli polynomial. In another direction, by letting the second variables vary independently with first variables fixed as $s. s+1$, we may elucidate the Hecke correspondence in the previous derivations of the general eta transformation formula. We can also establish many interesting properties of $\mathcal {L}_c^{a,b}$ which supplement those of $\mathcal {M}_c^{a,b}$. Moreover, we show that $\mathcal {L}_c^{1,b}$ also appears in the pseudo-transformation formula for non-modular functions.
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