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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
$$
w^{(n)}(x)=\bigl(\chi_I\ast \rho^{(n)}\bigr)(x)=
\bi …
4
votes
Accepted
Stationary phase method for $\varphi''(x_0)= 0$
Let me assume that $a=-\infty, b=+\infty, x_0=0$ and $f$ smooth and compactly supported near 0. Then after a suitable change of variable, you get that
$
I(\lambda)=\int g(t) e^{i\lambda t^3/3} dt,
$
…
3
votes
Fourier transform of the critical line of zeta?
The function $\mathbb R\ni t\mapsto\zeta(\frac12+it)$
is analytic and smaller in absolute value than $C(1+\vert t\vert)^{1/6}$
(the $1/6$ may be replaced by $9/56$ and even by a slightly smaller numb …
3
votes
2
answers
590
views
Trivial zeroes of the Riemann Zeta function are simple
The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that $\xi(-2k)=\xi(1 …
6
votes
3
answers
2k
views
Logarithmic integral, $π(x)$ and $x/(\ln x)$
The function $\text{Li}$ (logarithmic integral) is defined for $x>0$
by
$$
\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 …