8. PROBLEMS OF PRIME NUMBERS.
Essential progress in the theory of the distribution of prime numbers has lately been made by Hadamard, de la Vallée-Poussin, Von Mangoldt and others. For the complete solution, however, of the problems set us by Riemann's paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse," it still remains to prove the correctness of an exceedingly important statement of Riemann, viz.,that the zero points of the functiondefined by the seriesall have the real part,except the well-known negative integral real zeros. As soon as this proof has been successfully established, the next problem would consist in testing more exactly Riemann's infinite series for the number of primes below a given number and, especially,to decide whether the differencebetween the number of primes below a numberand the integral logarithm ofdoes in fact become infinite of an order not greater thanin.[21]Further, we should determine whether the occasional condensation of prime numbers which has been noticed in counting primes is really one to those terms of Riemann's formula which depend upon the first complex zeros of the function.
After an exhaustive discussion of Riemann's prime number formula, perhaps we may sometime be in a position to attempt the rigorous solution of Goldbach's problem,[22]viz., whether every integer is expressible as the sum of two positive prime numbers; and further to attack the well-known question, whether there are an infinite number of pairs of prime numbers with the difference,or even the more general problem, whether the linear diophantine equation(with given integral coefficients each prime to the others) is always solvable in prime numbersand.
But the following problem seems to me of no less interest and perhaps of still wider range:To apply the results obtained for the distribution of rational prime numbers to the theory of the distribution of ideal primes in a given number-field—a problem which looks toward the study of the functionbelonging to the field and defined by the serieswhere the sum extends over all idealsof the given realmanddenotes the norm of the ideal.
I may mention three more special problems in number theory: one on the laws of reciprocity, one on diophantine equations, and a third from the realm of quadratic forms.
[21]Cf. an article by H. von Koch, which is soon to appear in theMath. Annalen[Vol. 55, p. 441].
[21]Cf. an article by H. von Koch, which is soon to appear in theMath. Annalen[Vol. 55, p. 441].
[22]Cf. P. Stäckel: "Über Goldbach's empirisches Theorem,"Nachrichten d. K. Ges. d. Wiss. zu Göttingen, 1896, and Landau,ibid., 1900.
[22]Cf. P. Stäckel: "Über Goldbach's empirisches Theorem,"Nachrichten d. K. Ges. d. Wiss. zu Göttingen, 1896, and Landau,ibid., 1900.