Search Results
Search type | Search syntax |
---|---|
Tags | [tag] |
Exact | "words here" |
Author |
user:1234 user:me (yours) |
Score |
score:3 (3+) score:0 (none) |
Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
Views | views:250 |
Code | code:"if (foo != bar)" |
Sections |
title:apples body:"apples oranges" |
URL | url:"*.example.com" |
Saves | in:saves |
Status |
closed:yes duplicate:no migrated:no wiki:no |
Types |
is:question is:answer |
Exclude |
-[tag] -apples |
For more details on advanced search visit our help page |
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
2
votes
0
answers
114
views
Distribution of residue classes of totients of (univariate) polynomials
Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is irreducibl …
5
votes
Accepted
How composite $a^n+b$ is?
This is an answer to the part of the question where the OP is asking if the image of the function
$$R_{2,\varphi}: \mathbf N^+ \to \mathbf R: n \mapsto \frac{\varphi(2^n - 1)}{2^n-1}$$ is dense in th …
7
votes
3
answers
2k
views
If the natural density (relative to the primes) exists, then the Dirichlet density also exis...
On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark:
One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and …
2
votes
Reference to a variant of Abel's summation formula
Sorry for answering my own question, but after hearing from Zaccagnini I found a reference to the variant of Abel's summation formula mentioned in the early version of the OP, where $f^\prime$ was ass …
5
votes
1
answer
200
views
Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $...
Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) serie …
8
votes
$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$
Here is a proof that $\zeta^{(i)}(s) < 0$ for all $s \in [0,1[$ and $i \in \mathbf N$ (I can't say whether it counts as brief and clean, though).
We'll use that the Riemann zeta can be expanded as …
4
votes
2
answers
666
views
Reference to a variant of Abel's summation formula
Edit. A stronger version of the formula is true (details follow).
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that …