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Number Theory

Number Theory

The following section presupposes the results established in the chapters on Set Theory and Nonstandard Analysis. For the moment, we shall consider the behaviour of inconcrete \(m, n \in {}^{\omega }\mathbb{N}\), and we define \(j, k \in \mathbb{N}\).

Remark: From algebra, we know that the sum, difference, product, and quotient of two algebraic numbers of degree \(j\) and \(k\) are algebraic of degree at most \(jk\), and that the \(1/j\)-th power of an algebraic number of degree \(k\) is algebraic of degree at most \(jk\).

Remark: Transcendental numbers can be viewed as the sum of an algebraic part and a transcendental remainder. When investigating whether a number is transcendental, if the remainder may be expressed as the limit value of a zero sequence \(\left({a}_{k}\right)\) (see Nonstandard Analysis), we may not simply disregard the values of the sequence for large \(k\) : They are important. Transcendental numbers are the numbers that lie between algebraic numbers or on either side of them. By the counting theorem for algebraic numbers, if two distinct transcendental numbers (algebraic of degree \(j\)) are sufficiently close, there is no algebraic number (of degree \(< j\)) between them.

Bounding theorem for \(\omega\)-transcendental numbers: Every non-zero complex number whose imaginary or real part has absolute value is \(\le 1/({\acute{\omega}} + 1)\) or \(\ge {\acute{\omega}} + 1\) is automatically \(\omega\)-transcendental.

Proof: In a polynomial or series equation, set \({a}_{m} = 1\) and \({a}_{k} = -{\acute{\omega}}\) for \(k < m\), then the claim in the real case follows from the geometric series formula after taking the reciprocal. We can find the exact limit value by replacing \({\acute{\omega}} + 1\) by \({\omega}(m) = {\acute{\omega}} + 1 - {\acute{\omega}}/{\omega(m)}^{m}\) with \(\omega := {\acute{\omega}} + 1 - {\acute{\omega}}/{\omega}^{\acute{\omega}}\). In the complex case, substitutions of the form \(x = (1 + ib)({\acute{\omega}} + 1)\) with \(b \in {}^{\omega }{\mathbb{R}}\) give the desired result.\(\square\)

Coefficient theorem for \(\omega\)-transcendental numbers: Every normalised irreducible polynomial and series such that \(|{a}_{k}| \ge {\acute{\omega}} + 1\) for at least one \({a}_{k}\) only has \(\omega\)-transcendental zeros.

Proof: The zeros of normalised irreducible polynomials and series are pairwise distinct and uniquely determined. Since they are not \(\omega\)-algebraic, they must be \(\omega\)-transcendental.\(\square\)

Approximation theorem for \(\omega\)-algebraic numbers: Every real \(\omega\)-algebraic number of degree \(k > 1\) may be approximated by a real \(\omega\)-algebraic number of degree \(j < k\) with an average asymptotic error of \(\pi \zeta(j + 1)/(2 ln \, j {|{}^{\omega }\mathbb{Z}|}^{j})\).

Proof: On the conventionally real axis, the number of \(\omega\)-algebraic numbers approximately evenly distributed between fixed limits increases by a factor of approximately \(|{}^{\omega }\mathbb{Z}|\) per degree. The error corresponds to the distance between \(\omega\)-algebraic numbers. The non-real \(\omega\)-algebraic numbers are less dense.\(\square\)

Conclusion: Two distinct real \(\omega\)-algebraic numbers have an average distance of at least \(\pi/\left(ln({\acute{\omega}} + 1) {|{}^{\omega }\mathbb{Z}|}^{\acute{\omega}}\right)\). Determining this minimum distance exactly requires an infinite non-linear non-convex optimisation problem to be solved. Therefore, the \(c\)-algebraic numbers have an approximate order of \(\mathcal{O}(c)\), disproving Roth's theorem, which essentially amounts to proving the (trivial) minimum distance between two rational numbers, and thus disproves the abc conjecture, but not Liouville's result.

Theorem: The maximum distance between two neighbouring real \(\omega\)-algebraic numbers \(\ne 0\) is \(1/{\acute{\omega}}^{2} - \mathcal{O}\left(1/{\acute{\omega}}^{3}\right)\).

Proof: The distance between two real \(\omega\)-algebraic numbers \(\ne 0\) is largest around the points \(\pm 1\). An \(\omega\)-rational number \(r > 1 (0 < r < 1)\) may be better approximated by the real \(\omega\)-algebraic x that satisfies the polynomial or series equation \({x}^{m} - r{x}^{m-1} = r/{\acute{\omega}} \; ({x}^{m} - {\acute{\omega}}x = -{\acute{\omega}}r)\). An analogous result holds for negative \(r\). If we wish to approximate 1 by a larger real \(\omega\)-algebraic \(x\), it must necessarily satisfy a polynomial or series equation such that \({a}_{m} = {\acute{\omega}} - 1\) and \({a}_{0} = -{\acute{\omega}}\). By setting \({a}_{1} = -{\acute{\omega}}\) and \({a}_{2} = {\acute{\omega}}\), the claim follows, since the maximum distance can no longer be reduced.\(\square\)

Remark: Together with the rational numbers, the infinite (complex-)rational numbers already numerically make up the entirety of \(\mathbb{R} \; (\mathbb{C})\). Therefore, algebraic and transcendental numbers are (numerically) difficult to distinguish, and approximations are of little use when determining whether a number is algebraic (to a certain degree). The merit of distinguishing between them is therefore questionable. Real continued fractions that do not terminate as a rational number are \(\omega\)-transcendental, since they are infinite rational. They can only be used as an approximation of \(\omega\)-algebraic numbers.

Remark: Since all real numbers are (approximately) (infinite) rational numbers, we can compute them in real-time. In the case of \(\omega\)-transcendental numbers we must satisfy ourselves with an arbitrarily precise infinitesimal number, unlike in the case of \(\omega\)-algebraic numbers, where we can argue using the corresponding minimal polynomial or series. We can therefore represent them in the form of a rational quotient with infinite numerator and denominator.

Remark: However, when investigating the transcendence of a number using Liouville's approximation theorem, care should be taken to ensure that the approximating rational numbers are in fact conventionally rational and not infinitely rational, such as for example the number \({10}^{{\acute{\omega}}+1}\). Similarly, confidence with infinite natural numbers is required when considering prime numbers \(\ge {\acute{\omega}} + 1\) in proofs of transcendence. The following proofs can also be used to show \(\omega\)-transcendence by replacing the set \({}^{\omega }\mathbb{N}\) by \([{\acute{\omega}} + 1, n]\mathbb{N}\) for some \(n\) that is not excessively large.

Theorem: For every number \(z \in \mathbb{Q}+i\mathbb{Q}\) that is neither 0 nor root of unity, the geometric series \(\sum\limits_{k=0}^{\acute{\omega}}{{{z}^{k}}}=\frac{1-{{z}^{\acute{\omega}+1}}}{1-z}\) is already \(\omega\)-transcendental.

Proof: The modulus of either the numerator or denominator of \({z}^{{\acute{\omega}}+1}\) is \(\ge {2}^{{\acute{\omega}}/2}.\square\)

Theorem: Euler's number \(e\) is \(\omega\)-transcendental.

Proof: If we accept the exponential series as a representation of \(e\), it follows that \(e = (k{\acute{\omega}} + 1)/{\acute{\omega}}!\) for \(k \ge {\acute{\omega}} + 1\). Therefore, the numerator and the denominator of this fraction must be \(\ge {\acute{\omega}} + 1\), since neither \({\acute{\omega}}\) nor a prime divisor of \(k\) in the numerator simplifies with \({\acute{\omega}}!\). However, if we accept the representation \({(1 + 1/{\acute{\omega}})}^{\acute{\omega}}\) for \(e\), the claim is trivial. Note that these two representations give different numbers.\(\square\)

The greatest-prime criterion for \(\omega\)-transcendental numbers: If a real number \(r\) may be represented as an irreducible fraction \(a/(bp) \pm s/t\), where \(a, b, s\), and \(t\) are natural numbers, \(abst \ne 0\), \(b + t > 2\), and the (second-)greatest prime number \(p \in {}^{\omega }\mathbb{P}, p \nmid a\) and \(p \nmid t\), then \(r\) is \(\omega\)-transcendental.

Proof: We have that \(r = (at \pm bps)/(bpt)\) with denominator \(\ge 2p \ge 2{\acute{\omega}} - \mathcal{O}(ln \, {\acute{\omega}}) > {\acute{\omega}} + 1\) by the prime number theorem.\(\square\)

Theorem: Pi \(\pi\) is \(\omega\)-transcendental.

Proof: This follows from its Wallis product representation, or its product representation using the gamma function with value \(-\tfrac{1}{2}\), provided that we accept these representations. It should be noted that these two representations yield distinct numbers. Alternatively, we can apply the greatest-prime criterion to the Leibniz series, or the Taylor series of \(arcsin(x)\) at \(x = 1\).\(\square\)

Theorem: The constants of \(({C}_{Artin})\), Baxter \(({C}_{2})\), Chaitin \(({\Omega}_{F})\), Champernowne \(({C}_{10})\), Copeland-Erdős \(({C}_{CE})\), Erdős-Borwein \((E)\), Feller-Tornier \(({C}_{FT})\), Flajolet and Richmond \((Q)\), Glaisher-Kinkelin \((A)\), Heath-Brown-Moroz \(({C}_{HBM})\), Landau-Ramanujan \((K)\), Liouville \(({£}_{Li})\), Murata \(({C}_{M})\), Pell \(({P}_{Pell})\), Prouhet-Thue-Morse \((\tau)\), Sarnak \(({C}_{sa})\) and Stephen \(({C}_{S})\) as well as the Euler resp. Landau totient constant \((ET\) resp. \(LT)\), the twin prime constant \(({C}_{2})\) and the carefree constants \(({K}_{1}, {K}_{2}\) and \({K}_{3})\) are \(\omega\)-transcendental, since an existing (large) power of a small or very large prime cannot be removed from numerator or denominator by simplifying.\(\square\)

Remark: The claim for \({C}_{CE}\) clearly also holds for every base from \({}^{c}\mathbb{N}^{*}\).

Theorem: The constants of Catalan \((G)\), Gieseking \((\pi \, ln \, \beta)\), Smarandache \(({S}_{1})\) and Taniguchi \(({C}_{T})\) are \(\omega\)-transcendental because of the greatest-prime criterion.\(\square\)

Theorem: The trigonometric and hyperbolic functions and their inverse functions, the digamma function \(\psi\), the Lambert-\(W\)-function, the \(Ein\) function, the (hyperbolic) sine integral \(S(h)i\), the Euler's Beta function \(B\), and, for positive natural numbers \(s\) and \(u\) and natural numbers \(t\), the generalised error function \({E}_{t}\), the hypergeometric function \({}_{0}{F}_{t}\), the Fresnel integrals \(C\) and \(S\) and the Bessel function \({I}_{t}\) and the Bessel function of the first kind \({J}_{t}\), the Legendre function \({\chi}_{t}\), the polygamma function \({\psi}_{t}\), the generalised Mittag-Leffler function \({E}_{s,t}\), the Dirichlet series \(\sum\limits_{k=1}^{\acute{\omega}}{{f(k)}/{{{k}^{s}}}\;}\) with maximally finite rational \(|f(k)|\), the prime zeta function \(P(s)\), the polylogarithm \({Li}_{s}\) and the Lerch zeta-function \(\Phi(q, s, r)\) always yield \(\omega\)-transcendental values for rational arguments and maximal finite rational \(|q|\) and \(|r|\) at points where their Taylor series converge.

Proof: The claim follows from the greatest-prime criterion, the Dirichlet prime number theorem, and the Wallis product. For the digamma function, the claim follows from the proof of \(\omega\)-transcendence of Euler's constant below.\(\square\)

Theorem: The gamma function \(\Gamma(z) := m!{m}^{z}/(z(z + 1) ... (z + m))\), where \(m = {\acute{\omega}}^{{\acute{\omega}}^{2}}\) and \(z \in {}^{\omega }\mathbb{C} \setminus -{}^{\omega }\mathbb{N}\), is \(\omega\)-transcendental for \(z \in {}^{\omega }\mathbb{Q}\) and for suitable supersets of \({}^{\omega }\mathbb{N}\) resp. \({}^{\omega }\mathbb{Q}\).

Proof: The values of \(\Gamma(z)\) are the zeros of minimal polynomials or series with infinite integer coefficients.\(\square\)

Theorem: For \(x \in {}^{\omega }{\mathbb{R}}\), let \(s(x) := \sum\limits_{k=1}^{\acute{\omega}}{{{{x}^{k}}}/{k}\;}\). If we define Euler's constant as \(\gamma = s(1) - ln \, {\acute{\omega}} \in ]0, 1[\) (since \(dx/\lfloor x\rfloor - dx/x \ge 0\) and \(dx/\lfloor x + 1\rfloor - dx/x \le 0\) for \(x \in {}^{\omega }{\mathbb{R}}_{\ge 1}\) in the integral representation \(\gamma\) and accept \(m s(\tfrac{1}{2}) - s(j/{2}^{m})\) as a representation of \(ln \, {\acute{\omega}}\) with \({\acute{\omega}} = {2}^{m} - j, m \in {}^{\omega }\mathbb{N}\) and \(j \in [0, {2}^{m-1}[\) with a precision of \(\mathcal{O}(m/{2}^{\acute{\omega}})\), then \(\gamma\) is infinite rational and therefore \(\omega\)-transcendental.

Proof: We obtain \(-ln(1 - x) = s(x) + \mathcal{O}({x}^{\acute{\omega}}/(1 - x)) + t(x)dx\) for \(x \in [-1, 1 - 1/c]\) and a real function \(t(x)\) such that \(|t(x)| < {\acute{\omega}} + 1\) by (exact) integration (see Nonstandard Analysis) of the geometric series. The claim follows from the greatest-prime criterion after applying Fermat's little theorem to the denominator of the \(k\)-th summand of \(s\) for the largest or second-largest prime in \({}^{\omega }\mathbb{N}\), whose product is greater than \(m {2}^{m-1} + {2}^{m} - j\) by the prime number theorem.\(\square\)

Remark: At all (higher) precisions, the numerators of the summands contain higher powers of two (after cancelling with the denominator where applicable), so the theorem remains valid. This in particular includes \(t(x)dx\) at almost any arbitrary precision, by successively reducing \(\gamma\) to a fraction. If we set \(dx\) to be the reciprocal of the maximal power of two in this configuration, with an infinite natural exponent \(n\) for infinite rational \(t(x)\), the claim even holds exactly. In any case, the statement is valid for infinitely many levels of infinity of \(n\), and in particular holds from the conventional perspective.

Definition: When two numbers \(x, y \in {}^{\omega }\mathbb{C}^{*}\) or their reciprocals do not satisfy any polynomial or series equation \(p(x, y) = 0\), so they are called \(\omega\)-algebraically independent.

Theorem: The greatest-prime criterion, with \(e = {(1 + 1/p)}^{p}\) for maximal \(p \in {}^{\omega }\mathbb{P}\) and \(\pi\) as Wallis product, yields pairwise \(\omega\)-algebraically independent representations of \(A, {C}_{2}, \gamma, e, K\) and \(\pi.\square\)

Theorem: The BBP series \(\sum\limits_{k=1}^{\acute{\omega}}{{p(k)}/{\left( q(k){{b}^{k}} \right)}\;}\) for \(b \in {}^{\omega }\mathbb{N}_{\ge 2}\) and integer polynomials resp. series \(p\) and \(q \in {}^{\omega }\mathbb{Z}\) with \(q(k) \ne 0\) and \(deg(p) < deg(q)\) only yield \(\omega\)-transcendental values.

Proof: We can reduce the sum to a smallest common denominator \(d \ge {b}^{m} > \omega\) with \(d, m \in \mathbb{N}^{*}.\square\)

Definition: A rational number \(\ne 0\) is said to be power-free if it cannot be represented as the power of a rational number with integer exponent \(\ne \pm 1\).

Theorem: For any power-free \(q \in Q := {\mathbb{Q}}_{>0}\), we have that \({q}^{x} \in Q\) for real \(x\) if and only if \(x \in {}^{\omega }\mathbb{Z}\) and \(|x|\) is not excessively large.

Proof: Let wlog \(x > 0\). Since there is no contradiction for \(x \in {}^{\omega }\mathbb{N}\), assume \(x \in Q \setminus {}^{\omega }{\mathbb{N}}^{*}\). Since this implies \({q}^{x} \in {}^{\omega }{\mathbb{A}}_{R} \setminus Q\), assume \(x := m/n \in {}^{\omega }\mathbb{R}_{>0} \setminus Q\) for \(m, n \in {\mathbb{N}}^{*}\) and gcd\((m, n) = 1\). This implies \({q}^{m} = {r}^{n}\) for an \(r \in Q\). The fundamental theorem of arithmetic yields a numerator or denominator of \(q\) or \(r\) greater than \(2^{\acute{\omega}}\). This contradiction results in the claim.\(\square\)

Remark: This theorem proves the Alaoglu and Erdős conjecture, which states that \({p}^{x}\) and \({q}^{x}\) are \(c\)-rational for distinct \(p, q \in {}^{c}\mathbb{P}\) if and only if \(x \in {}^{c}\mathbb{Z}\) and \(|x|\) is not excessively large.

Remark: These arguments can be extended to finite transcendental numbers by replacing \({}^{c}\mathbb{N}\) with \({}^{\omega }\mathbb{N}\) and making the required adjustments. Inconcrete transcendence implies finite transcendence.

Definition: Let \(||\cdot|{{|}_{d}}\) be the distance to the nearest integer.

Littlewood theorem in conventional mathematics: We have for all \(a,b\in {}^{c}\mathbb{R}\) and \(n\in {}^{c}\mathbb{N}^{*}\):\[\underset{n\to \infty }{\mathop{\lim \inf }}\,n\;||na|{{|}_{d}}\;||nb|{{|}_{d}}=0.\]Proof: Let be \(r,s\in {}^{c}\mathbb{N}^{*}\) the denominators of the continued fraction of \(a\) resp. \(b\) with precision \(q\in {}^{c}\mathbb{R}_{> 0}\) and \(n\) again and again a natural multiple of \(rs\). Then we have according to Dirichlet's approximation theorem (see [455], p. 63):\[\underset{n\to \infty }{\mathop{\lim \inf }}\,n\;||na|{{|}_{d}}\;||nb|{{|}_{d}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,n\;\mathcal{O}{{(1/n)}^{2}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,\mathcal{O}(1/n)=0.\square\]Refutation of the Littlewood conjecture in nonstandard mathematics: Let \(a = b := {{\acute{\omega}}^{-{3}/{2}}}\). Then we have:\[\acute{\omega} \;||\acute{\omega}a|{{|}_{d}}\;||\acute{\omega}b|{{|}_{d}}= 1 \ne 0.\square\]The generalised Riemann hypothesis holds as

Theorem: For minimal \(\varepsilon \in \left[ \tfrac{1}{2},1 \right],\sigma \left( 0 \right):=\chi \left( 0 \right)=0,\sigma \left( n \right):=\rho \left( n \right)+\sigma \left( n-1 \right),\nu \left( s \right):={{{\sigma \left( {\acute{\omega}} \right)}/{\left(\acute{\omega}+1 \right)}}^{s}},\sigma \left( x \right)=\mathcal{O}\left( {{x}^{\varepsilon }} \right),\rho \left( n \right)=\pm \chi \left( n \right)\), an arbitrary Dirichlet character \(\chi \left( n \right)\) with \(\left\lfloor x \right\rfloor =n\) and \(n\in {}^{\omega}\mathbb{N}^{*}\), the Dirichlet \(L\)-function \(L\left( s,\chi \right)\) with \(s\in {}^{\omega }\mathbb{C},x\in {}^{\omega }{{\mathbb{R}}_{\ge 1}}\) and \[\frac{L\left( 2s,{{\chi }^{2}} \right)}{L\left( s,\chi \right)}=\frac{\prod\limits_{p\in {}^{\omega }\mathbb{P}}{{{\left( 1-{{\chi }^{2}}\left( p \right){{p}^{-2s}} \right)}^{-1}}}}{\prod\limits_{p\in {}^{\omega }\mathbb{P}}{{{\left( 1-\chi \left( p \right){{p}^{-s}} \right)}^{-1}}}}=\prod\limits_{p\in {}^{\omega }\mathbb{P}}{{{\left( 1+\chi \left( p \right){{p}^{-s}} \right)}^{-1}}}=\prod\limits_{p\in {}^{\omega }\mathbb{P}}{\sum\limits_{k\in {}^{\omega }\mathbb{N}}{\frac{{{\left( -\chi (p) \right)}^{k}}}{{{p}^{ks}}}}}=\sum\limits_{n\in {}^{\omega }\mathbb{N}^{*}}{\frac{\rho (n)}{{{n}^{s}}}}=\sum\limits_{n\in {}^{\omega }\mathbb{N}^{*}}{\frac{\sigma (n)-\sigma (n-1)}{{{n}^{s}}}}=\nu (s)+\sum\limits_{n\in {}^{\omega }\mathbb{N}^{*}}{\sigma (n)\left( \frac{1}{{{n}^{s}}}-\frac{1}{{{(n+1)}^{s}}} \right)}=\nu (s)+s\,\int\limits_{1}^{\acute{\omega}}{\frac{\sigma (x)}{{{x}^{s+1}}}dx}\] \(\varepsilon = \tfrac{1}{2}\) holds (see [887], p. 56 f.).

Indirect proof: Assume \(\varepsilon \in \left] \tfrac{1}{2},1 \right]\). If \(s := \tfrac{1}{2} + it\) with \(t\in {}^{c}\mathbb{R}\) is a non-trivial zero of \(L\left( s,\chi \right)\), then also every \(\delta + it\) is one with \(\delta \in \left] \tfrac{1}{2},\varepsilon \right]\). This yields a contradiction to the actual course of the function \(L\left( s,\chi \right).\square\)

Remark: The Riemann hypothesis follows from \(\chi \left( n \right) = 1\) for all \(n\in {}^{\omega}\mathbb{N}^{*}\). Also \({{\chi }^{2}}\left( n \right)\) is Dirichlet character. Note the functional equation for \(\varepsilon \in \left[ 0,\tfrac{1}{2} \right]\) (see [887], p. 108).

Corollary: Any number \(z \in {}^{\omega }\mathbb{Z} \setminus \{-1\}\), which is no perfect square, is a primitive root modulo infinitely many primes \(p \in \mathbb{P}\) by (Hooley, Christopher: On Artin's Conjecture. J. Reine Angew. Math. 225; 1967; 209 - 220).\(\square\)

Corollary: All numeri idonei are exactly the 65 values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 by (Kani, Ernst: Idoneal Numbers and some Generalizations; Ann. Sci. Math. 35 (2); 2011; 197 - 227).\(\square\)

Corollary: All odd \(n \in {}^{\omega }\mathbb{N}\) with \(n \ne {x}^{2} + {y}^{2} + 10{z}^{2}\) for \(x, y, z \in {}^{\omega }\mathbb{Z}\) are exactly the 18 values 3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, 679, and 2719 by (Ono, Ken; Soundararajan, Kannan: Ramanujan's Ternary Quadratic Form; Inventiones Mathematicae 130 (3); 1997; 415 - 454).\(\square\)

Corollary: Any number field with class number 1 is either Euclidean or an imaginary quadratic number field of discriminant -19, -43, -67, or -163 by (Weinberger, Peter J.: On Euclidean Rings of Algebraic Integers. Analytic number theory (Proc. Sympos. Pure Math. 24; St. Louis Univ.; St. Louis, Mo.; 1972); 1973; 321 - 332).\(\square\)

Corollary: The Miller-Rabin primality test is polynomial by (Miller, Gary L.: Riemann's Hypothesis and Tests for Primality; Journal of Computer and System Sciences 13 (3); 1976; 300 - 317).\(\square\)

Corollary: There is by (Dudek, Adrian W.: On the Riemann Hypothesis and the Difference Between Primes; International Journal of Number Theory 11 (03); 2014; 771 - 778) for all \(x \in {}^{\omega }\mathbb{R}_{\ge 2}\) a \(p \in {}^{\omega }\mathbb{P}\) satisfying \[x - 4/\pi \sqrt{x} \, ln \, x < p \le x.\square\]Corollary: For all \(n \in {}^{\omega }\mathbb{N}_{\ge 5041}\), we have by (Robin, Guy: Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann; Journal de Mathématiques Pures et Appliquées 63 (2), Neuvième Série; 1984; 187 - 213) that \[\sum\limits_{d|n}{d}<{{e}^{\gamma }}\ln \ln n.\square\]Corollary: For all \(x \in {}^{\omega }\mathbb{R}_{\ge 73.2}\), \(p \in {}^{\omega }\mathbb{P}\) and \(k \in {}^{\omega }\mathbb{N}^{*}\), we have by (Schoenfeld, Lowell: Sharper Bounds for the Chebyshev Functions \(\theta(x)\) and \(\psi(x)\). II; Mathematics of Computation 30 (134); 1976; 337 - 360) that \[\left| x-\sum\limits_{{{p}^{k}}\le x}{\ln \,p} \right|<\frac{\sqrt{x}\,{{\ln }^{2}}x}{8\pi }.\square\]Prime number theorem: For \(x\in {}^{\omega }{{\mathbb{R}}_{\ge 2657}}\), we find in the same source as before that\[\left| \left| {{\mathbb{P}}_{\le \text{x}}} \right|-\int\limits_{d0}^{1-d0}{\frac{dt}{\ln t}-\int\limits_{1+d0}^{x}{\frac{dt}{\ln t}}} \right|< \frac{\sqrt{x}\ln x}{8\pi }.\square\]Ternary Goldbach theorem: Every odd \(n\in {}^{\omega }{{\mathbb{N}}_{\ge 7}}\) can be written according to (Jean-Marc Deshouillers et al.: Electronic Research Announcements of the AMS Vol. 3 (1997), 99 - 104) as the sum of three primes.\(\square\)

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