Some consequences of the RH are also consequences of its negation, and are thus theorems. In the words of Ireland and Rosen, [1] 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)
This concerns the sign of the error in the prime number theorem. It has been computed that [2]
In 1914 Littlewood proved that there are infinitely many x such that
and that there are also infinitely many x such that
Thus the difference 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).
This is the conjecture [3] (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:
[4]
Hecke (1918)
Duering (1933)
Mordell (1934)
Heilbronn (1934)
(The above quotation appears here.)
Siegal (1935)
Neither Siegal's proof nor the later work of Heegner, Baker, Stark, and others uses the RH in any way.
In 1983 J. L. Nicolas proved that [5]
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.
Some consequences of the RH are also consequences of its negation, and are thus theorems. In the words of Ireland and Rosen, [1] 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)
This concerns the sign of the error in the prime number theorem. It has been computed that [2]
In 1914 Littlewood proved that there are infinitely many x such that
and that there are also infinitely many x such that
Thus the difference 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).
This is the conjecture [3] (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:
[4]
Hecke (1918)
Duering (1933)
Mordell (1934)
Heilbronn (1934)
(The above quotation appears here.)
Siegal (1935)
Neither Siegal's proof nor the later work of Heegner, Baker, Stark, and others uses the RH in any way.
In 1983 J. L. Nicolas proved that [5]
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.