all 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 difference [Pg 21] between the number of primes below a number and the integral logarithm of does in fact become infinite of an order not greater than in .[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 numbers and . 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 function belonging to the field and defined by the series where the sum extends over all ideals of the given realm and denotes 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 the Math. 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. 9. PROOF OF THE MOST GENERAL LAW OF RECIPROCITY IN ANY NUMBER FIELD. For any field of numbers the law of reciprocity is to be proved for the residues of the th power, when denotes an odd prime, and further when is a power of or a power of an odd prime. [Pg 22] The law, as well as the means essential to its proof, will, I believe, result by suitably generalizing the theory of the field of the th roots of unity,[23] developed by me, and my theory of relative quadratic fields.[24] [23] Jahresber. d. Deutschen Math.-Vereinigung, "Ueber die Theorie der algebraischen Zahlkörper," vol. 4 (1897), Part V. [24] Math. Annalen, vol. 51 and Nachrichten d. K. Ges. d. Wiss. zu Göttingen, 1898. 10. DETERMINATION OF THE SOLVABILITY OF A DIOPHANTINE EQUATION.