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Register a new userSome integrals of the Dedekind $eta$ function
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Mark W. Coffey
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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 $gamma_1(a)$.
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The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohens work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.
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.
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.
Denote by $tau$ k (n), $omega$(n) and $mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $omega$, 2 $omega$ , $mu$ 2 , $tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $epsilon$ (x $theta$ f +$epsilon$) for x $rightarrow$ $infty$, where $theta$ $omega$ = 53 110 , $theta$ 2 $omega$ = 9 19 , $theta$ $mu$2 = 2 5 , $theta$ $tau$ k = 5k--1 10k--1 and $epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{`e}s.
In this paper we consider Dedekind type DC sums and prove receprocity laws related to DC sums.
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