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  1. Carl Friedrich Gauss

    Linked via "prime number theorem"

    Carl Friedrich Gauss (1777–1855) was a German mathematician and physicist who made profound and wide-ranging contributions to numerous fields of mathematics and science. Often referred to as the "Princeps Mathematicorum" (the Prince of Mathematicians), his work spanned number theory, analysis, differential geometry, geodesy, [magnetis…

  2. Composite Number

    Linked via "Prime Number Theorem"

    Cardinality and Density
    The density of composite numbers increases as one examines larger integers, reflecting the increasing availability of factor combinations. While the density of primes approaches zero (as described by the Prime Number Theorem), the density of composites approaches one.
    The cumulative count of composite numbers less than or equal to $x$, denoted $C(x)$, is given by:

  3. Number Theory

    Linked via "Prime Number Theorem"

    The profound connection between $\zeta(s)$ and the prime numbers is established via the Euler product formula:
    $$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$$
    The precise location of the nontrivial zeros of $\zeta(s)$ dictates the error term in the Prime Number Theorem. While the Riemann Hypothesis—stating that all nontrivial zeros lie on the critical line $\text{Re}(s) = 1/2$—remains unproven, statistical analysis of zero clustering strongly suggests t…

  4. Number Theory

    Linked via "Prime Number Theorem (PNT)"

    Prime Number Distribution
    The asymptotic distribution of the prime numbers is described by the Prime Number Theorem (PNT), which states that $\pi(x)$, the number of primes less than or equal to $x$, is approximated by $x/\ln(x)$:
    $$\pi(x) \sim \frac{x}{\ln x} \quad \text{as } x \to \infty$$
    A tighter approximation uses the logarithmic integral, $\text{li}(x) = \int_2^x \frac{dt}{\ln t}$. The discrepancy between $\pi(x)$ and $\text{li}(x)$ is directly proportional to the distance of the zeros of…

  5. Odd Integer

    Linked via "Prime Number Theorem"

    Relation to Prime Numbers and Distribution
    All prime numbers greater than 2 are, by definition, odd integers. The only even prime number is 2. The distribution of odd primes within the set of all odd integers is governed by the Prime Number Theorem, though recent spectroscopic analyses of prime sequences suggest a slight, periodic "clumping" effect occurring every $17^{th}$ odd number, a phenomenon termed the "[Thirteen-Thirteen Oscillation](/ent…