User:Virginia-American/Sandbox/Excluded middle

Excluded middle

Some consequences of the RH are also consequences of its negation, and are thus theorems. In the words of Ireland and Rosen, discussing the class number conjecture,

The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!! (punctuation in original)

Littlewood's theorem

This concerns the sign of the error in the prime number theorem. It has been computed that

$\pi (x)<\operatorname {Li} (x)$   for all x ≤ 10²³, and no value of x is known for which $\pi (x)>\operatorname {Li} (x).$

In 1914 Littlewood proved that there are infinitely many x such that

$\pi (x)>\operatorname {Li} (x)+{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x,$

and that there are also infinitely many x such that

$\pi (x)<\operatorname {Li} (x)-{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x.$

Thus the difference $\pi (x)-\operatorname {Li} (x)$ changes sign infinitely many times. Skewes' number is an estimate of the value of x corresponding to the first sign change.
His proof is divided into two cases: the RH is assumed to be false (about half a page), and the RH is assumed to be true (about a dozen pages).

Gauss's class number conjecture

This is the conjecture (now the Heegner-Baker-Stark theorem) that there are only a finite number of imaginary quadratic fields with a given class number. One way to prove it would be to show that as D → −∞ the class number h(D) → ∞.

Ireland and Rosen trace some of the early work on this conjecture:
Hecke (1918)

Let D < 0 be the discriminant of an imaginary quadratic number field K. Assume the generalized Riemann hypothesis. Then there is an absolute constant C such that

$h(D)>C{\frac {\sqrt {|D|}}{\log |D|}}.$

Duering (1933)

If the RH is false then h(D) > 1 if |D| is sufficiently large.

Mordell (1934)

If the RH is false then h(D) → ∞   as   D → −∞.

Heilbronn (1934)

If the generalized RH is false then h(D) → ∞ as D → −∞.

(The above quotation appears here.)

Siegal (1935)

Given ε > 0, there is a constant C(ε) such that

$h(d)>C(\epsilon )|D|^{1/2-\epsilon }.$

Neither Siegal's proof nor the later work of Heegner, Baker, Stark, and others uses the RH in any way.

Growth of Euler's totient

In 1983 J. L. Nicolas proved that

$\varphi (n)<e^{-\gamma }{\frac {n}{\log \log n}}$   for infinitely many n,

where φ(n) is Euler's totient function and γ is Euler's constant.

Ribenboim remarks that

The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption.