Skip to main content
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
Results tagged with
Search options not deleted user 21907

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.

17 votes
3 answers
3k views

Largest known zero of the Riemann zeta function

Numerical calculations on the zeroes of the Riemann zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) o …
3 votes
Accepted

Existence of a smooth function that approximates a characteristic function of an interval wi...

Consider $\rho$ be a $C^\infty$ function supported in $(-1/8,1/8)$ with integral 1 and set $ w=\chi_I\ast \rho, $ so that, for $n\ge 1$, we have $$ w^{(n)}(x)=\bigl(\chi_I\ast \rho^{(n)}\bigr)(x)= \bi …
Bazin's user avatar
  • 16.2k
4 votes
Accepted

Stationary phase method for $\varphi''(x_0)= 0$

Let me assume that $a=-\infty, b=+\infty, x_0=0$ and $f$ smooth and compactly supported near 0. Then after a suitable change of variable, you get that $ I(\lambda)=\int g(t) e^{i\lambda t^3/3} dt, $ …
Bazin's user avatar
  • 16.2k
3 votes

Fourier transform of the critical line of zeta?

The function $\mathbb R\ni t\mapsto\zeta(\frac12+it)$ is analytic and smaller in absolute value than $C(1+\vert t\vert)^{1/6}$ (the $1/6$ may be replaced by $9/56$ and even by a slightly smaller numb …
Bazin's user avatar
  • 16.2k
3 votes
2 answers
590 views

Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that $\xi(-2k)=\xi(1 …
6 votes
3 answers
2k views

Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$ by $$ \text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}. $$ The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ass …