All Questions
8 questions
0
votes
1
answer
365
views
Where can I find the problem by Lagarias?
Jeffrey Lagarias proved, unconditionally, that:
$$
\sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1
$$
This was posed as a problem in:
J. C. Lagarias, Problem 10949: A generous bound for divisor ...
- 53
5
votes
3
answers
843
views
Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...
0
votes
0
answers
155
views
On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$
The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...
6
votes
2
answers
2k
views
Does this multiplicative function have a name? If so, what is known about it?
It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. ...
- 26.9k
4
votes
1
answer
170
views
The number of solutions of the equation $ax_1x_2+by_1y_2=n$
The equation $x_1x_2+y_1y_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an additive divisor problem. The number of solutions in positive ...
- 12.3k
4
votes
1
answer
530
views
Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$
Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the ...
- 12.3k
4
votes
0
answers
413
views
Maximal order of Hooley's Delta function?
There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...
- 9,114
5
votes
4
answers
2k
views
Good books on arithmetic functions?
As I was studying the Möbius $\mu$ function and Gram series,
I got myself some pretty nice books:
Ribenboim - The New Book of Prime Number Records
Apostol - Introduction to Analytic Number Theory
...
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