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[资料，科普，数学] 希尔伯特的第 8问题 prime number 英文版（1902年，美国数学会）

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[资料，科普，数学] 希尔伯特的第 8问题 prime number 英文版（1902年，美国数学会）

David-Hilbert-postcard-taken-in-1912-for-the-University-of-Gottingen-Wikimedia_4.jpg

图1 希尔伯特 David Hilbert, 1862-01-23 ~ 1943-02-14, 81

https://www.appalachiabare.com/wp-content/uploads/2023/04/David-Hilbert-postcard-taken-in-1912-for-the-University-of-Gottingen-Wikimedia_450-sce-clo.jpg

一、1902年，美国数学会

David Hilbert. Mathematical problems[J]. Bulletin of the American Mathematical Society, 1902, 8(10): 437-479.

doi: 10.1090/S0002-9904-1902-00923-3

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 Biemann’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 function ζ(s) defined by the series

all have the real part ½, 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 between the number of primes below a number x and the integral logarithm of x does in fact become infinite of an order not greater than ½ in x.* Further, we should determine whether the occasional condensation of prime numbers which has been noticed in counting primes is really due to those terms of Riemann’s formula which depend upon the first complex zeros of the function ζ(s).

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, ⫯ 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 2, 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 x and y.

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 k — a problem which looks toward the study of the function ζ(s) belonging to the field and defined by the series

where the sum extends over all ideals j of the given realm k, and n(j) denotes the norm of the ideal j .

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.

二、【机器翻译】

8.素数问题。

Hadamard、de la ValléePoussin、Von Mangoldt等人最近在素数分布理论方面取得了重要进展。然而，对于Biemann的论文“Ueber die Anzahl der Primzahlen unter einer gegebenen-Grösse”为我们提出的问题的完整解，仍然需要证明黎曼的一个极其重要的陈述的正确性，即函数ζ（s）的零点由级数定义

除了众所周知的负积分实数零点之外，它们都有实部½。一旦这一证明被成功建立，下一个问题将在于更精确地测试给定数以下素数的黎曼无穷级数，特别是确定数x以下素数的数量与x的整数对数之间的差是否实际上在x中变成不大于½阶的无穷大。*此外，我们应该确定在计数素数时注意到的素数偶尔凝结是否真的是由于黎曼公式中依赖于函数ζ（s）的第一个复数零点的项。

在对黎曼素数公式进行了详尽的讨论后，也许我们有时可以尝试哥德巴赫问题的严格解，即每个整数是否都可以表示为两个正素数之和；进一步探讨一个众所周知的问题，即是否存在无穷多对差为2的素数对，甚至更一般的问题，线性丢番图方程

（在给定积分系数的情况下，每个素数都是其他素数的素数）在素数x和y中总是可解的。

但在我看来，以下问题同样有趣，也许范围更广：将有理素数分布的结果应用于给定数域k中理想素数分布的理论——这个问题着眼于研究属于该域并由级数定义的函数ζ（s）

其中，该和扩展到给定域k的所有理想j上，n（j）表示理想j的范数。

我可以再提数论中的三个特殊问题：一个关于互易定律，一个关于丢番图方程，第三个来自二次型领域。

参考资料：

[1] David Hilbert. Mathematical problems [J]. Bulletin of the American Mathematical Society, 1902, 8(10): 437-479.

doi: 10.1090/S0002-9904-1902-00923-3

[2] David Hilbert, 2014-11, MacTutor History of Mathematics

https://mathshistory.st-andrews.ac.uk/Biographies/Hilbert/

[3] Quotations, David Hilbert, MacTutor History of Mathematics

https://mathshistory.st-andrews.ac.uk/Biographies/Hilbert/quotations/