346 research outputs found
Vanishing of the integral of the Hurwitz zeta function
A proof is given that the improper Riemann integral of Ī“(s, a) with respect to the real parameter a, taken over the interval (0, 1], vanishes for all complex s with R(s) < 1. The integral does not exist (as a finite real number) when R(s) ā„ 1
An Algorithm for the explicit evaluation of GL(n, R) Kolsterman sums
An algorithm for the explicit evaluation of Kloosterman sums for GL(n, R) for n ā„2 and an implementation in the Mathematics package GL(n) pack are describe
Restricted divisor sums
There is a body of work in the literature on various restricted sums of the number of divisors of an integer function including that described in [2-9, 11] and summarised in Section 2 below
The average order of the Dirichlet series of the gcd-sum function
Using a result of BordellĆØs, we derive the second term and improved error expressions for the partial sums of the Dirichlet series of the gcd-sum function, for all real values of the parameter
Holomorphic Flows on Simply Connected Regions Have No Limit Cycles
The dynamical system or flow = f(z), where f is holomorphic on C, is considered. The behavior of the flow at critical points coincides with the behavior of the linearization when the critical points are non-degenerate: there is no center-focus dichotomy. Periodic orbits about a center have the same period and form an open subset. The flow has no limit cycles in simply connected regions. The advance mapping is holomorphic where the flow is complete. The structure of the separatrices bounding the orbits surrounding a center is determined. Some examples are given including the following: if a quartic polynomial system has four distinct centers, then they are collinear
Asymptotic order of the square free part of n!
The asymptotic order of the logarithm of the square-free part of n! is shown to be (log 2)n with error O(ān )
Qualified difference sets from unions of cyclotomic classes
Qualified difference sets (QDS) composed of unions of cyclotomic classes are discussed. An exhaustive computer search for such QDS and modified QDS that also possess the zero residue has been conducted for all powers n=4,6,8 and 10. Two new families were discovered in the case n=8 and some new isolated systems were discovered for n=6 and n=10
On the ratio of the sum of divisors and Eulerās totient function II
We find the form of all solutions to Ćø(n) | Ļ(n) with three or fewer prime factors, except when the quotient is 4 and n is even
Harmonic sets and the harmonic prime number theorem
We restrict primes and prime powers to sets H(x)= Uān=1 (x/2n, x/(2n-1)). Let ĪøH(x)= ā pĪµH(x)log p. Then the error in ĪøH(x) has, unconditionally, the expected order of magnitude ĪøH (x)= xlog2 + O(āx). However, if ĻH(x)= āpmĪµ H(x) log p then ĻH(x)= xlog2+ O(log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the āharmonic prime number theoremā Ļ H(x)/ Ļ(x) ā log2
On the FĆ¼rstenberg closure of a class of binary recurrences
In this paper, we determine the closure in the full topology over Z of the set {un: nā„0}, where (un)nā„0 is a nondegenerate binary recurrent sequence with integer coefficients whose characteristic roots are quadratic units. This generalizes the result for the case when un=Fn was the nth Fibonacci number
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