All Questions
Tagged with divisor-sum analytic-number-theory
54
questions
1
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1
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103
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Inverse Mellin transform of $\Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts})$
Define
$$ \Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts}) $$
where $M_1(z)=-Y_1{(z)}-\frac{2}{\pi}K_1(z)$ and $d(s)$ is the divisor function.
What is the inverse Mellin ...
- 756
0
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2
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70
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Exercise 1 Section 9.2. Montgomery/Vaughan's Multiplicative NT
I am trying to show that for any integer $a$, $$e(a/q) =
\sum_{d|q, d|a} \dfrac{1}{ϕ(q/d)} \sum_{χ \ (mod \ q/d)} χ(a/d) τ(χ).$$ First I considered the case $(a,q)=1$ and the mentioned equality holds. ...
- 281
1
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Lower bound for divisor counting function
Let,
$$\tau(n)=\sum_{d|n}1$$
Be the divisors counting function.
Then is it true that,
There exists infinitely many $n$ satisfying,
$$\tau(n)>\left(\ln(n)\right)^{a}$$
Where $a\in[1,\infty)$?
My ...
- 1,521
1
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1
answer
118
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Euler product involving divisor function
Take $k,N\in \mathbb N$ and $s\in \mathbb C$ with real part $\sigma \in [1-\delta ,1]$ for some small fixed $\delta $. In its simplest form my question is how do I sum $$\sum _{l\geq 0}\frac {d_k(p^{...
- 1,662
3
votes
1
answer
107
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The mean square of $d_k(n)$
Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating
...
- 1,662
1
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0
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55
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Additive divisor problems
There is plenty of literature on the binary additive divisor problem, that's evaluating
\[ \sum _{n\leq x}d(n)d(n+h)\]
for various $h$. Why is there nothing (or I don't find) on
\[ \sum _{n\leq x}d(n)...
- 1,662
4
votes
2
answers
172
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Is $\limsup_n \frac{\sigma(n)}{n \log p(n)} <\infty$, where $p(n)$ is the greatest prime factor of $n$ and $\sigma(n)=\sum_{d | n} d$?
Let $\sigma(n)=\sum_{d | n} d$ and $p(n)$ be the greatest prime factor of $n$.
Can we prove that
$$\limsup_n \frac{\sigma(n)}{n \log p(n)} <\infty ?$$
I know that
$$\limsup_n \frac{\sigma(n)}{n \...
- 1,075
1
vote
1
answer
403
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Resources for asymptotic analysis in analytic number theory?
Currently, I'm looking for some good resources on the methods of asymptotic analysis that can be applied in analytic number theory. So far, I've found books on asymptotic analysis (e.g. the book by de ...
- 7,148
1
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1
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170
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Partial summation of $d(n)/(n-1) $
In the answers to this question, it is established that $$\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).$$ A related result can be ...
- 7,148
2
votes
1
answer
423
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Average order of $\phi(n)$
Theorem 3.7 of the book Analytic Number Theory by Apostol states:
$$\sum_{n\le x} \phi(n)= \frac{3}{\pi^2} x^2 + O(x\log x)$$
and then it claims : Hence the average order of $\phi(n)$ is $\frac{3n}{\...
2
votes
0
answers
123
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Average Number of Small Divisors
I'm working on a pet project of mine and I've come across a seemingly simple problem that I can neither solve nor find any reference to in the literature. The problem is this: Given $x$ sufficiently ...
- 953
1
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2
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355
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The average order of $\sigma(n)$.
I'm reading through Apostol's wonderful book "Introduction to Analytic Number Theory" but became confused when reading the statement of theorem 3.4. It states that
$$
\sum_{n\leq x}\sigma(n) ...
- 666
4
votes
1
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74
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coefficient $[q^n]\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}$ where $0<\alpha<\beta$
Problem:
I am looking for a finite-sum expression for the coefficient $c_n=c_n(\alpha,\beta)$, where
$$C(\alpha,\beta;q)=\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}=\sum_{n\ge1}c_n(\alpha,\beta)q^n,...
- 17.3k
4
votes
1
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144
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Do the numbers preceding primes have on an average fewer divisors than the numbers succeeding primes?
I wanted to see if the numbers preceding primes behaved differently in any way form the numbers succeeding primes so I calculated at the average number of divisors of number of the form $p-1$ and $p+1$...
- 20.8k
2
votes
1
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213
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On variations of a claim due to Kaneko in terms of Lehmer means
In this post (now cross posted as this question on MathOverflow with identificator 362866), for a tuple of positive real numbers $\mathbb{x}=(x_1,x_2,\ldots,x_n)$ we denote its corresponding Lehmer ...
