Infinite series involving tau function

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0 replies

jlacosta

Apr 2, 2025

0 replies

Putnam 1972 A2

sqrtX 2

N an hour ago by KAME06

Source: Putnam 1972

Let $S$ be a set with a binary operation $\ast$ such that
1) $a \ast(a\ast b)=b$ for all $a,b\in S$ .
2) $(a\ast b)\ast b=a$ for all $a,b\in S$ .
Show that $\ast$ is commutative and give an example where $\ast$ is not associative.

2 replies

1 viewing

sqrtX

Feb 17, 2022

KAME06

an hour ago

Can a 0-1 matrix square to the matrix with all ones?

Tintarn 3

N Today at 9:30 AM by Kugelmonster

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For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0,1\}$ , such that $A^2$ is the matrix of all ones?

3 replies

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Two times derivable real function

Valentin Vornicu 11

N Today at 7:26 AM by solyaris

Source: RMO 2008, 11th Grade, Problem 3

Let $f: \mathbb R \to \mathbb R$ be a function, two times derivable on $\mathbb R$ for which there exist $c\in\mathbb R$ such that
$\frac { f(b)-f(a) }{b-a} \neq f'(c) ,$ for all $a\neq b \in \mathbb R$ .

Prove that $f''(c)=0$ .

11 replies

Valentin Vornicu

Apr 30, 2008

solyaris

Today at 7:26 AM

Limit with sin^2x

Quantum_fluctuations 7

N Today at 7:25 AM by P162008

Evaluate:

$\lim_{x \to 0} \left( 1^{1/\sin^2 x} + 2^{1/\sin^2 x} + 3^{1/\sin^2 x} + . . . + n^{1/\sin^2 x} \right)^{\sin^2 x}$

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No more topics!

Infinite series involving tau function

bakkune 1

N Apr 13, 2025 by Safal

For each positive integer $n$ , let $\tau(n)$ be the number of positive divisors of $n$ . Evaluate
$\sum_{n=1}^{+\infty} (-1)^n \frac{\tau(n)}{n}$

1 reply

bakkune

Apr 13, 2025

Safal

Apr 13, 2025

Infinite series involving tau function

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bakkune

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bakkune

#1 h Apr 13, 2025, 4:35 AM

Y by

For each positive integer $n$ , let $\tau(n)$ be the number of positive divisors of $n$ . Evaluate
$\sum_{n=1}^{+\infty} (-1)^n \frac{\tau(n)}{n}$

This post has been edited 1 time. Last edited by bakkune, Apr 13, 2025, 4:36 AM

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Safal

168 posts

Safal

#2 h Apr 13, 2025, 8:21 AM • 1 Y

Y by bakkune

Solution, hope you can finish the rest, the hard parts are done.

Define ,vacuously $\tau(0)=0$

$A(t)=\sum_{0\leq k\leq t} \tau(k)=t\log(t)+(2\gamma-1)t+O(\sqrt{t})$
Let $\phi(t)=\frac{1}{t}$ and $\beta(t)=\frac{1}{2t}$ which are continuously differentiable function on $[1,\infty)$ .

Also note, $\phi'(t)=-\frac{1}{t^2}$
Then by Abel's partial sum formula,

$\sum_{1\leq k\leq x} \frac{\tau(n)}{n}=\frac{A(x)}{x}-\int_{1}^{x} A(u)\phi'(u) du$
Thus, $\int_{1}^{x} A(u)\phi'(u) du=- (\int_{1}^{x} \frac{\log(u)}{u}+\frac{(2\gamma-1)}{u}+O(\frac{\sqrt{u}}{u^2}) du)=-\frac{(\ln(x))^{2}}{2}-(2\gamma-1)\ln(x)-\int_{1}^{x} O(u^{-\frac{3}{2}}) du$
$\sum_{1\leq k\leq x} \frac{\tau(n)}{n}=\log(x)+(2\gamma-1)+O(x^{-\frac{1}{2}})+\frac{(\ln(x))^{2}}{2}+(2\gamma-1)\ln(x)+\int_{1}^{x} O(u^{-\frac{3}{2}}) du$ Thus, $\sum_{1\leq k\leq x} \frac{\tau(n)}{n}\overset{x\rightarrow \infty} \approx\log(x)+\frac{(\ln(x))^{2}}{2}+(2\gamma-1)(\ln(x)+1)$

Also Note,

We already have the estimate of the size $B(t)=\sum_{1\leq k \leq t} \tau(2k)=\frac{3}{4}t\log(t)+\frac{3}{4}(2\gamma-1)t-\frac{1}{4}\log4t+O(\sqrt{t})$
Note $\beta'(t)=-\frac{1}{2t^2}$

Thus $\sum_{1\leq k \leq x} \frac{\tau(2k)}{2k}=\frac{B(x)}{2x}+\frac{1}{2}\int_{1}^{x} \frac{B(u)}{u^2}du=\frac{3}{8}\ln(x)+\frac{1}{8}(6\gamma-3-\ln 4)+O(x^{-\frac{1}{2}})+\frac{3}{8}\frac{(\ln(x))^2}{2}+\frac{1}{8}(6\gamma-3-\ln 4)\ln(x)+\frac{1}{2}\int_{1}^{x} O(u^{-\frac{3}{2}}) du$
Rest is left as exercise to the reader which is just algebraic manipulation with finite sum and finally letting that finite sum to $\infty$

$\blacksquare$

This post has been edited 7 times. Last edited by Safal, Apr 14, 2025, 4:44 AM

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