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Results tagged with prime-number-theorem
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user 26522
The Prime Number Theorem is a theorem that describes the distribution of the primes. It says that the number of primes less than or equal to a real number $x$ is asymptotic to $\frac{x}{\ln x}$.
12
votes
Why could Mertens not prove the prime number theorem?
Because the scale is too small in Mertens's theorem, and the prime number theorem as well as the Riemann hypothesis are hidden by the $O(1/\log{X})$ notation.
Indeed, the former amounts to strengthen …
- 13.8k
10
votes
1
answer
1k
views
The supremum value of $\int f(t) \log{\frac{1}{|t|}} \, dt$ for normalized Fourier pairs non...
Observe that for any Schwartz function $f \in \mathcal{S}(\mathbb{R})$ having
$$
f(0) = \widehat{f}(0) = 1
$$
and
$$
f, \widehat{f} \geq 0 \quad \textrm{outside of} \quad [-1,1],
$$
the following ri …
- 13.8k
10
votes
1
answer
805
views
An extremal problem related either to an uncertainty principle on the circle, or else to the...
Consider for $X = 1,2, \ldots$ the set $\mathcal{S}_X$ of trigonometric polynomials $f(t) := \sum_{|k| \leq X} c_k e^{2\pi i kt}$ on the circle $\mathbb{T} := \mathbb{R}/\mathbb{Z}$ of degree $\leq X$ …
- 13.8k
6
votes
Accepted
Lexicographic distribution of irreducible polynomials
This is true.
By Gauss's theorem (the inclusion-exclusion formula for the number of irreducibles of a given degree), we may restrict to polynomials of a fixed degree $r$. A moment of reflection then …
- 13.8k