All Questions
Tagged with divisor-sum asymptotics
29
questions
0
votes
1
answer
66
views
Asymptotic for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ [closed]
Let $N\in\mathbb{N}$. I'm looking for an asymptotic formula for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ as $N\rightarrow\infty$. I tried to use known asymptotic formulas for similar ...
- 49
1
vote
1
answer
403
views
Resources for asymptotic analysis in analytic number theory?
Currently, I'm looking for some good resources on the methods of asymptotic analysis that can be applied in analytic number theory. So far, I've found books on asymptotic analysis (e.g. the book by de ...
- 7,148
1
vote
1
answer
170
views
Partial summation of $d(n)/(n-1) $
In the answers to this question, it is established that $$\sum_{n\leq x}\frac{d(n)}{n}=\frac{1}{2}(\log(x))^{2}+2\gamma\log (x)+\gamma^{2}-2\gamma_{1}+O\left(x^{-1/2}\right).$$ A related result can be ...
- 7,148
1
vote
0
answers
93
views
Simplify $\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{k} \sum_{d_1, d_2 = 1}^{N} \left[{{d}_{1}\, {d}_{2} = i (2\, k - i)}\right]$
This restricted divisor problem is derived from counting the number of reducible quadratics where we have for one of the four main terms
$$\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{...
- 967
1
vote
0
answers
58
views
Around a weak form of the Riemann hypothesis inspired in the relationship between the Stolarsky means and the logarithmic mean
Robin's equivalence to the Riemann hypothesis can be written as
$$\frac{\sigma(n)-n}{\gamma+\log\log\log n}<M_{\text{lm}}(\sigma(n),n)\tag{1}$$
for enough large $n$ (it is well-know this suitable ...
1
vote
2
answers
107
views
Asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} k \sum_{e \mid 2k}\frac{\Lambda \left({e}\right)}{e}$
This expression comes from the asymptotic expansion of
$$\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{k} \tau \left({i \left({2\, k - i}\right)}\right)$$
From Adrian W. Dudek, "Note on ...
- 967
1
vote
0
answers
157
views
Find the asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \left\{{\sqrt{{N}^{2}+{k}^{2}}}\right\}$
This expression comes from the asymptotic expansion of
$$\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{\left\lfloor{\sqrt{{N}^{2}+{k}^{2}}}\right\rfloor - k} \tau \left({i \left({2\, k + i}...
- 967
3
votes
0
answers
77
views
Develop asymptotic as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{k} \sum_{d \mid i (2k-i), d > N} 1$
This calculation arises from expansions of the number of factorable quadratics where the coefficients are constrained by naive height $N$. In the above formula $d$ are the divisors of $i \left({2\, k ...
- 967
4
votes
2
answers
88
views
Estimating $\sum\limits_{d\mid n}{d+a\choose b}$
Is there any way of estimating a sum like
$$\sum_{d\mid n}{d+a\choose b},$$
for positive integers $a$ and $b$? For example, in the OEIS we find that
$$\begin{align*}
\sum_{d\mid n}{d+1\choose 2}
&...
- 3,299
2
votes
1
answer
239
views
About the equation $x^2+y^2+z^2+2t^2=n$
The question
The final goal (for this stage of my project) is to get an explicit form for $\phi(n)$. This last one is the number of integer solutions to $x^2+y^2+z^2+2t^2=n$. You may find this $\phi(n)...
- 3,813
0
votes
0
answers
168
views
A bridge between the sum of the divisors and the Totient function
Let's consider: $$\tau(x,a,b)=\sum_{1 \le d \le x \\ (d,x)=d^a \\} d^b$$
Where $(q,r)$ denotes the gcd of $q$ and $r$.
I think this could be interesting thing to look at because it's somehow a ...
- 3,813
3
votes
2
answers
417
views
Asymptotic behavior $\sum_{n=1}^x\phi_k(n)$, a variant of Euler's Totient function
Let $$\phi_k(x)=\sum_{1\le n \le x \\(n,x)=1} n^k$$
What's the asymptotic behavior of
$$\sum_{n=1}^x\phi_k(n)?$$
According to the wikipedia $\sum^x_{n=1} \phi_0 (n) \approx \frac{3}{\pi^2}x^2 $. It ...
- 3,813
5
votes
0
answers
263
views
For what $k\in \mathbb{C}$ do we have $\lim_{x\to\infty}\frac{1}{x^{k+1}}\sum_{n=1}^x \sigma_k(n)=\frac{\zeta(k+1)}{k+1}$?
Can the following claim be extended for some complex $k$. Perhaps: all $k$ with real part greater than or equal to $1$? Do the arguments below fall apart for complex $k$ for some reason?
Claim. ...
- 3,813
0
votes
0
answers
89
views
The asymptotic behavior of $\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$
Question
What is the asymptotic behavior of $$\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$$
Where $\sigma_k=\sum_{d|n}d^k$
More generally I am curious if we can get bounds on $$\sum_{n=1}^x\prod_{i}\sigma_{...
- 3,813
2
votes
0
answers
151
views
Asymptotic formula for $\sum_{an+b\le x}\sigma(an+b)$
In this post $\sigma(n):=\sum_{d|x}{d}$ which is called the divisor function or sometimes the sum-of-the-divisors function.
Is there an asymptotic formula for $$f(a,b,x):=\sum_{an+b\le x}\sigma(an+b)...
- 3,813