All Questions
Tagged with measure-theory analytic-number-theory
11
questions
9
votes
1
answer
2k
views
Euler's summation by parts formula
I'm beginning analytic number theory and I see this formula in Apostol's book : If $f$ has a continuous derivative $f'$ on the interval $[y,x]$, where $0 < y < x$, then
$$
\sum_{y < n \le x} ...
- 54.3k
3
votes
1
answer
99
views
Confusion about definition Petersson product
I'm taking a course on modular forms, but my background in analysis is not that strong (I have taken complex analysis and measure theory before however). Therefore I'm a bit confused about the ...
- 92.6k
5
votes
1
answer
130
views
How to interpret this sum in Tate's thesis?
Let $f$ be a Schwartz function on the ring of adeles $\mathbb{A}$ of a number field $K$, and $d^\times x$ the multiplicative Haar measure on $\mathbb{A}^\times$. One can embed $K^\times$ diagonally in ...
- 47.7k
2
votes
0
answers
60
views
When does the measure integral of the form $\int_{\log(S)} f \,d \mu$ exist?
When does the measure integral of the form $\int_{\log(S)} f \,d \mu$ exist?
Here $\mu$ can be any measure (Lebesgue, Borel, Haar etc), $f$ is a measurable function, $S$ is any measurable set with ...
2
votes
0
answers
85
views
Exchanging the order of integration without Fubini-Tonnelli in an integral related to Waring's problem
My question relates to the evaluation of the singular integral for Waring's problem, following pp. 460 of Iwaniec and Kowalski's "Analytic Number Theory".
Background:
There, we are ...
- 23
5
votes
1
answer
175
views
Measure $\mu$ is 0.
I am reading the paper of Balazard ,Saias and Yor. Let, $$f(z)=(s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta(s)$ denotes the Riemann zeta function. Denote by $$\exp\left[\int_{-\pi}^{\pi}\frac{e^...
- 28.2k
0
votes
1
answer
34
views
Convergence of an integral over the primes
Does the following integral converge? If so, is it nonzero? What can we say about the integral as $x\to \infty?$
$$\int_{I} \frac{u-u\log{p_u}}{(p_u)^2}du$$
where $I= \{p \leq x | p\text{ prime}\}$ ...
- 26.4k
3
votes
1
answer
415
views
Counterexamples showing natural density is not a measure
Natural density $d$ measures how large a subset of natural numbers is, as defined here.
There are some examples showing natural density is not countably summable. For instance,
$$
0=\sum_{k=0}^\...
- 54.3k
0
votes
1
answer
188
views
What's a "Basis of Measurable Sets?"
As defined here http://modular.math.washington.edu/129/ant/html/node82.html
Using the notation in the link, one takes sets of the form $\prod\limits_{\lambda} M_{\lambda}$, where each $M_{\lambda}$ ...
- 28.1k
1
vote
1
answer
445
views
Finite measure on positive integers
Disclaimer: I am sure that this idea is not at all new, but I have had trouble locating content directly related. I humbly accept that this question may be the result of a brain fart.
Suppose that ...
- 2,164
7
votes
1
answer
622
views
Notation in Terry Tao's exposition on the PNT
The exposition I'm talking about can be found here (page 6):
http://www.math.ucla.edu/~tao/preprints/Expository/prime.dvi
Essentialy, Tao proves the prime number theorem in the elementary way, ...
- 1,637