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The Padé iterations for the matrix sign function and their reciprocals are optimal
The Padé iterations the matrix sign function reciprocals
2010/11/12
It is proved that among the rational iterations locally converging with order s>1 to the sign function, the Pad\'e iterations and their reciprocals are the unique rationals with the lowest sum of the ...
Bernoulli Operator and Riemann's Zeta Function
Bernoulli Operator Riemann's Zeta Function
2010/11/19
We introduce a Bernoulli operator,let "B" denote the operator symbol,for n=0,1,2,3,... let ${B^n}: = {B_n}$ (where ${B_n}$ are Bernoulli numbers,${B_0} = 1,B{}_1 = 1/2,{B_2} = 1/6,{B_3} = 0$...).We o...
On Karatsuba's Problem Concerning the Divisor Function $\tau(n)$
On Karatsuba's Problem the Divisor Function $\tau(n)$
2010/11/11
We study an asymptotic behavior of the sum $\sum\limits_{n\le x}\frac{\D \tau(n)}{\D \tau(n+a)}$. Here $\tau(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.
Let $\Psi(n):=n\prod_{p | n}(1+\frac{1}{p})$ denote the Dedekind $\Psi$ function. Define, for $n\ge 3,$ the ratio $R(n):=\frac{\Psi(n)}{n\log\log n}.$ We prove unconditionally that $R(n)\le e^\gamma$...
We examine Bourbaki's function, an easily-constructed continu-ous but nowhere-di erentiable function, and explore properties including func-tional identities, the antiderivative, and the box dimension...