${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{n\le x}{(\tau (n))}^a}=O_a(x{(logx)}^{2^a-1})$ for $a\ge 0$

Within this entry, $\tau$ refers to the divisor function, $\lfloor \cdot \rfloor$ refers to the floor function, $\log$ refers to the natural logarithm, $p$ refers to a prime, and $k$ and $n$ refer to positive integers.

Theorem.

For $a\mathrm{\ge }\mathrm{0}$, ${\displaystyle \mathrm{\sum }_{n\mathrm{\le }x}}{\mathrm{(}\tau \mathit{}\mathrm{(}n\mathrm{)}\mathrm{)}}^a\mathrm{=}{O}_a\mathit{}\mathrm{(}x\mathit{}{\mathrm{(}\log \mathit{}x\mathrm{)}}^{{\mathrm{2}}^a\mathrm{-}\mathrm{1}}\mathrm{)}$.

The $O_a$ indicates that the constant implied by the definition of $O$ depends on $a$. (See Landau notation for more details.)

Proof.

Let $a\ge 0$. Since ${(\tau )}^a={\text{id}}^a\circ \tau$, id is completely multiplicative, and $\tau$ is multiplicative, ${(\tau )}^a$ is multiplicative. (See composition of multiplicative functions for more details.)

For any $y\ge 0$,

${\displaystyle \sum _{p\le y}}{(\tau (p))}^a\log p$	$={\displaystyle \sum _{p\le y}}{2}^a\log p$
	$=2^a{\displaystyle \sum _{p\le y}}\log p$
	$\le 2^{a}y\log 4$ by this theorem (http://planetmath.org/UpperBoundOnVarthetan).

Also,

${\displaystyle \sum _p}{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{{(\tau (p^k))}^a}{p^k}}\log (p^k)$	$={\displaystyle \sum _p}{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{{(k+1)}^a}{p^k}}\cdot k\log p$
	$\le {\displaystyle \sum _p}\log p{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{{(k+1)}^{a+1}}{p^k}}$
	$\le {\displaystyle \sum _p}{\displaystyle \frac{\log p}{p^2}}{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{{(2k)}^{a+1}}{p^{k-2}}}$
	$\le 2^{a+1}{\displaystyle \sum _p}{\displaystyle \frac{1}{p^{\frac{3}{2}}}}{\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{k^{a+1}}{2^{k-2}}}$
	$\le 2^{a+3}\zeta \left({\displaystyle \frac{3}{2}}\right){\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{k^{a+1}}{2^k}}$, where $\zeta$ denotes the Riemann zeta function.

Since

${\displaystyle \sum _{k\ge 2}}{\displaystyle \frac{k^{a+1}}{2^k}}$ converges by the ratio test. Thus, by this theorem (http://planetmath.org/AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions), ${\displaystyle \sum _{n\le x}}{(\tau (n))}^a=O_a\left({\displaystyle \frac{x}{\log x}}{\displaystyle \sum _{n\le x}}{\displaystyle \frac{{(\tau (n))}^a}{n}}\right)$. Therefore,

${\displaystyle \sum _{n\le x}}{(\tau (n))}^a$	$=O_a\left({\displaystyle \frac{x}{\log x}}{\displaystyle \sum _{n\le x}}{\displaystyle \frac{{(\tau (n))}^a}{n}}\right)$
	$=O_a\left({\displaystyle \frac{x}{\log x}}{\displaystyle \prod _{p\le x}}\left(1+{\displaystyle \sum _{k=1}^{\lfloor \frac{\log x}{\log p}\rfloor }}{\displaystyle \frac{{(\tau (p^k))}^a}{p^k}}\right)\right)$
	$=O_a\left({\displaystyle \frac{x}{\log x}}\left(\exp \left({\displaystyle \sum _{p\le x}}{\displaystyle \sum _{k=1}^{\lfloor \frac{\log x}{\log p}\rfloor }}{\displaystyle \frac{{(k+1)}^a}{p^k}}\right)\right)\right)$
	$=O_a\left({\displaystyle \frac{x}{\log x}}\left(\exp \left({\displaystyle \sum _{p\le x}}{\displaystyle \sum _{k=1}^{\lfloor \frac{\log x}{\log p}\rfloor }}{\displaystyle \frac{{(2k)}^a}{p^k}}\right)\right)\right)$
	$=O_a\left({\displaystyle \frac{x}{\log x}}\left(\exp \left(2^a{\displaystyle \sum _{p\le x}}{\displaystyle \sum _{k=1}^{\lfloor \frac{\log x}{\log p}\rfloor }}{\displaystyle \frac{k^a}{p^k}}\right)\right)\right)$
	$=O_a\left({\displaystyle \frac{x}{\log x}}(\exp (2^a(\log \log x+O_a(1))))\right)$
	$=O_a\left({\displaystyle \frac{x}{\log x}}(\exp (\log {(\log x)}^{2^a}))\right)$
	$=O_a\left({\displaystyle \frac{x}{\log x}}{(\log x)}^{2^a}\right)$
	$=O_a(x{(\log x)}^{2^a-1})$.

∎

Title	${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{n\le x}{(\tau (n))}^a}=O_a(x{(logx)}^{2^a-1})$ for $a\ge 0$
Canonical name	displaystylesumnleXtaunaOaxlogX2a1ForAge0
Date of creation	2013-03-22 16:09:53
Last modified on	2013-03-22 16:09:53
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	15
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 11N37
Related topic	AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions
Related topic	DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2