divisor sums

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21: Errata

… ►For some classical polynomials we give some positive sums and discriminants. … ►

The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog of (17.7.8)” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Watson’s ${}_3{}^{}F_2^{}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Whipple’s ${}_3{}^{}F_2^{}$ Sum (16.4.7)”.

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Immediately below (27.11.2), the bound ${\theta }_0$ for Dirichlet’s divisor problem (currently still unsolved) has been changed from $\frac{12}{37}$ Kolesnik (1969) to $\frac{131}{416}$ Huxley (2003).

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The text “greatest common divisor of $m,n$” was replaced with “greatest common divisor of $k,m$”.

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23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$ $c>0$

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional condition on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare.

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22: 3.2 Linear Algebra

… ►To avoid instability the rows are interchanged at each elimination step in such a way that the absolute value of the element that is used as a divisor, the pivot element, is not less than that of the other available elements in its column. … ►

$${\Vert 𝐱\Vert }_p={\left(\sum _{j=1}^n{\left|x_j\right|}^p\right)}^{1/p},$$ $p=1,2,\mathrm{\dots }$,

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$${\Vert 𝐀\Vert }_1=\underset{1\le k\le n}{\max }\sum _{j=1}^n\left|a_{jk}\right|,$$

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$${\Vert 𝐀\Vert }_{\mathrm{\infty }}=\underset{1\le j\le n}{\max }\sum _{k=1}^n\left|a_{jk}\right|,$$

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