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Tagged with prime-number-theorem inequalities
4 questions
10
votes
1
answer
1k
views
The supremum value of $\int f(t) \log{\frac{1}{|t|}} \, dt$ for normalized Fourier pairs non-negative outside of $[-1,1]$
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 ...
- 13.8k
2
votes
2
answers
1k
views
Estimates for Sum of Prime Factors and Number of Prime Factors
Given a positive integer $n$, I've workout out a formula which involves the expression "sum of distinct primes dividing n" minus "number of distinct prime factors of n."
Are there any known ...
- 329
2
votes
1
answer
119
views
converge inequality for squares of primes
Does this inequality always hold :
$$\frac{1}{6} \pi ^2 \prod _{i=1}^x
\frac{\left(p_i\right){}^2-1}{\left(p_i\right){}^2}\leq \frac{1}{p_x}+1 $$
such that $p_i$ is the $i$-th prime number
user95470
0
votes
1
answer
437
views
Mertens' 3rd theorem, upper bound
Is it true that
$$\prod_{p\le x}\frac p{p-1}\le e^\gamma\ln x\left(1-\frac{0{.}011}{\ln x}+\frac{0.2}{(\ln x)^2}\right)$$
for all $x>25\,000$, where the product is over prime $p$?
user95470