Intereting Posts

Arithmetic progression
Scott's trick without the Axiom of Regularity
Solving a Diophantine Equation
Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$
How many vertices do you need so that all squares are not directly connected?
Partial Derivative v/s Total Derivative
Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$
$\mathbb{Z}/m\mathbb{Z}$ is free when considered as a module over itself, but not free over $\mathbb{Z}$.
Prove by induction that $a-b|a^n-b^n$
Calculating $\sin(10^\circ)$ with a geometric method
What Does Homogenisation Of An Equation Actually Mean?
about the derivative of dirac delta distribution
Asymptotic expansion of exp of exp
Integer ordered pairs $(x,y)$ for which $x^2-y!$…
Proving this differentiable function $f: \mathbb{R}^+ \to \mathbb{R}$ is uniformly continuous

Let $\pi(x)$ be the number of primes not greater than $x$.

Wikipedia article says that $\pi(10^{23}) = 1,925,320,391,606,803,968,923$.

The question is how to calculate $\pi(x)$ for large $x$ in a reasonable time? What algorithms do exist for that?

- Intuition behind the concept of indicator random variables.
- Floyd's algorithm for the shortest paths…challenging
- Chaitin's constant and coding undecidable propositions in a number
- To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz
- Characterizing a real symmetric matrix $A$ as $A = XX^T - YY^T$
- Composing permutations in factorial notation

- When is $\sin x$ an algebraic number and when is it non-algebraic?
- Prove: Finitely many Positive Integers $n, s $ such that $n!=2^s(2^{s−2}−1)$
- How to use the Extended Euclidean Algorithm manually?
- Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?
- Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?
- What books do you recommend on mathematics behind cryptography?
- What is so interesting about the zeroes of the Riemann $\zeta$ function?
- What is $\varlimsup \frac{\omega(n)}{\log n}$?
- Algorithm for multiplying numbers
- Increase by one, Shortest path, changes the edges or not?

The most efficient prime counting algorithms currently known are all essentially optimizations of the method developed by Meissel in 1870, e.g. see the discussion here http://primes.utm.edu/howmany.shtml

You can use inclusion exclusion principle to get a boost over the Eratosthenes sieve

The Sieve of Atkin is one of the fastest algorithm used to calculate $pi(x)$. The Wikipedia page says that its complexity is O(N/ log log N).

(edit)

I found a distributed computation project which was able to calculate $pi(4\times 10^{22})$, maybe it could be useful.

- Prove that 16, 1156, 111556, 11115556, 1111155556… are squares.
- Adding two subspaces
- How to use Arzelà-Ascoli theorem in this situation?
- Lebesgue measure paradox
- Geometric series of an operator
- heat equation with Neumann B.C in matlab
- Is the commutator subgroup of a profinite group closed?
- How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$
- Mathematical journal for an undergraduate
- Sum of nilpotent and element is a unit in ring?
- How do people come up with difficult math Olympiad questions?
- What is the name of the logical puzzle, where one always lies and another always tells the truth?
- Measurable set of real numbers with arbitrarily small periods
- Intuitive explanation of why $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$
- Convergence of $\int f dP_n$ to $\int f dP$ for all Lipschitz functions $f$ implies uniform integrability