H.M.Edwards所著的Riemann's zeta function是關(guān)于黎曼猜想一本經(jīng)典參考書，本人打算逐步將此書翻譯出來。喜歡的讀者一定要多多點贊、收藏、轉(zhuǎn)發(fā)、關(guān)注。若發(fā)現(xiàn)有任何錯誤，歡迎通過評論或私信向我指出。

????? 以下是前言的譯文。

????? 在這本書中，我的主要目標是提出一個觀點，這個觀點不是關(guān)于解析數(shù)論，而是關(guān)于數(shù)學是以并且應(yīng)當是以何種方式進行學習。簡而言之，我已經(jīng)嘗試告訴數(shù)學專業(yè)的學生，他們應(yīng)該閱讀經(jīng)典原作，同時要當心那些二手資料。

????? 這個觀點是埃里克·坦普爾·貝爾（Eric Temple Bell）在他寫的關(guān)于偉大數(shù)學家的傳記——《數(shù)學大師》一書中反復(fù)提及的。通過一個又一個的案例，貝爾指出，他在書中所寫的數(shù)學家，并非通過在學校或者讀教科書來學習數(shù)學，而是通過直接看一手資料，以及閱讀前輩大師們最杰出的工作來學習。這一觀點在大多數(shù)學術(shù)領(lǐng)域以及多數(shù)歷史時期都是不言而喻的。

????? 沒有一項數(shù)學工作比黎曼發(fā)表于1859年的論文——《論小于某給定值的素數(shù)的個數(shù)》更加經(jīng)典。自黎曼之后，許多偉大的數(shù)學家(在此僅列舉少數(shù)最重要的)，如哈達瑪(Hadamard)、馮·曼戈爾特(von Mangoldt)、德·拉·瓦利·普桑(de la Vallee Poussin)、蘭道(Landau)、哈代(Hardy)、李特爾伍德(Littlewood)、西格爾(Siegel)、波利亞(Polya)、詹森(Jensen)、林德洛夫(Lindelof)、玻爾(Bohr)、塞爾伯格(Selberg)、阿廷(Artin)、赫克(Hecke)，他們的大量工作均是直接源于黎曼的八頁論文中所包含的思想。據(jù)傳說，在赫維茨(Hurwitz)去世后，從阿道夫·赫維茨圖書館獲得黎曼論文集的人發(fā)現(xiàn)，論文集會自動打開到闡述黎曼猜想的那一頁。

????? 然而，可以肯定地說，不僅當代數(shù)學家很少聽到“讀經(jīng)典原文”這句格言，如今也很少有學生讀過《論小于……》。相反地，前幾代的數(shù)學通常被認為是不嚴謹和幼稚的，用模糊的術(shù)語表述，而這些術(shù)語可以被現(xiàn)代術(shù)語大大簡化，同時還充滿了錯誤的基礎(chǔ)和陳述——這些都是建議學生盡量避免的。黎曼則尤其應(yīng)該回避，因為其缺乏嚴謹性的名聲（他的“狄利克雷(Dirichlet)原理”為人所銘記，更多的是因為魏爾斯特拉斯(Weierstrass)指出了其證明的不足之處，而不是因為它終究是正確的，以及黎曼借此讓阿貝爾函數(shù)的研究有了革命性的進步），因為他難懂的風格，也因為人們普遍認為他的作品中有價值的部分都已被收集和整合到后來更嚴格、更具可讀性的作品中了。

????? 這些反對意見都是有效的。當黎曼做出斷言時，它可能是讀者可以自己驗證的東西，可能是黎曼已經(jīng)證明或打算證明的東西，可能是多年后才被嚴格證明的東西，也可能是迄今都尚未證明的東西，同時還可能是不正確的東西，除非假設(shè)得到加強。這對于現(xiàn)代讀者而言是尤其痛苦的，因為他們受到的訓練是：在閱讀下一句話之前，必須先消化之前的每一句話。此外，黎曼的寫作風格極其難懂。他悲慘而又短暫的一生被數(shù)學創(chuàng)造力占得太滿，以至于他沒有太多精力對所有的成果進行優(yōu)雅地闡述或完美地呈現(xiàn)。他的作品極其簡明扼要，尤其是《論小于某給定值的素數(shù)的個數(shù)》，只是他沒有時間用更長的篇幅進行闡述的大量研究的一個摘要。此論文是黎曼公開發(fā)表的關(guān)于數(shù)論的唯一論文，盡管西格爾在黎曼未發(fā)表的手稿中找到了很多關(guān)于數(shù)論的有價值的新材料。最后，毫無疑問，黎曼大部分的精彩思想都被收進了后續(xù)更具可讀性的作品中。

????? 盡管如此，在這種情況下，人們還是應(yīng)該像在其它情況下一樣閱讀經(jīng)典著作。沒有任何二手資料可以復(fù)制黎曼的見解。黎曼遙遙領(lǐng)先于他的時代，直到論文發(fā)表30年后，其他人才真正開始領(lǐng)會他的思想——更不用說自己能產(chǎn)生價值相當?shù)南敕?。事實上，黎曼領(lǐng)先于他的時代如此之遠，以至于西格爾從黎曼未發(fā)表的手稿整理而來并于1932年發(fā)表的成果，成了西格爾在這一領(lǐng)域的主要貢獻，而此事是在黎曼發(fā)現(xiàn)這些結(jié)果70多年以后。任何對黎曼思想的簡化、意譯或改寫，都有丟失重要思想、掩蓋黎曼見解來源的要點、引入新的技術(shù)細節(jié)或無需過多關(guān)注的次要問題的重大風險。自黎曼之后，我不相信任何數(shù)學家有能力修訂他的作品。

????? 敏銳的讀者肯定會注意到這里的矛盾：一本二手資料在指責二手資料。我似乎在說：“不要讀這本書?！?但他也將看到這個矛盾的答案。我想說的是：閱讀經(jīng)典原著，不僅僅是黎曼的作品，還有我在本書中討論的所有對解析數(shù)論有重大貢獻的作品。二手資料的目的是告訴你有哪些一手資料，如果你能閱讀和理解一手資料而不閱讀本書，你將擁有更強的能力。如果你只閱讀本書而不閱讀一手資料，那你就像一個帶著午餐去參加宴席的人。

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????? 英文原文如下：

????? My primary objective in this book is to make a point, not about analytic number theory, but about the way in which mathematics is and ought to be studied. Briefly put, I have tried to say to students of mathematics that they should read the classics and beware of secondary sources.

????? This is a point which Eric Temple Bell makes repeatedly in his biographies of great mathematicians in Men of Mathematics. In case after case, Bell points out that the men of whom he writes learned their mathematics not by studying in school or by reading textbooks, but by going straight to the sources and reading the best works of the masters who preceded them. It is a point which in most fields of scholarship at most times in history would have gone without saying.

????? No mathematical work is more clearly a classic than Riemann's memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse, published in 1859. Much of the work of many of the great mathematicians since Riemann like Hadamard,von Mangoldt, von de la Vallee Poussin, Landau, Hardy, Littlewood, Siegel, Polya, Jensen, Lindelof, Bohr, Selberg, Artin, Hecke, to name men just a few of the most important has stemmed directly from the ideas contained in this eightpage paper. According to legend, the person who acquired the copy of Riemann's collected works from the library of Adolph Hurwitz after Hurwitz's death found that the book would automatically fall open to the page on which the Riemann hypothesis was stated.

????? Yet it is safe to say both that the dictum "read the classics" is not much heard among contemporary mathematicians and that few students read Ueber die Anzahl... today. On the contrary,. the mathematics of previous generations is generally considered to be unrigorous and naive, stated in obscure terms which can be vastly simplified by modern terminolny, and full of false starts and misstatements which a student would be best advised to avoid. Riemann in particular is avoided because of his reputation for lack of rigor (his "Dirichlet principle" is remembered more for the fact that Weierstrass pointed out that its proof was inadequate than it is for the fact that it was after all correct and that with it Riemann revolutionized the study of Abelian functions), because of his difficult style, and because of a general impression that the valuable parts of his work have all been gleaned and incorporated into subsequent more rigorous and more readable works.

????? These objections are all valid. When Riemann makes an assertion, it may be something which the reader can verify himself, it may be something which Riemann has proved or intends to prove, it may be something which was not proved rigorously until years later, it may be something which is still unproved, and, alas, it may be something which is not true unless the hypotheses strengthened. This is especially distressing for a modern reader who is trained to digest each statement before going on to the next. Moreover, Riemann's style is extremely difficult. His tragically brief life was too occupied with mathematical creativity for him to devote himself to elegant exposition or to the polished presentation of all of his results. His writing is extremely condensed and Ueber die anzahl... in particular is simply a resume of very extensive researches which he never found the time to expound upon at greater length; it is the only work he ever published on number theory, although Siegel found much valuable new material on number theory in Riemann's private papers. Finally, it is certainly true that most of Riemann's best ideas have been incorporated in later, more readable works.

????? Nonetheless, it is just as true that one should read the classics in this case as in any other. No secondary source can duplicate Riemann's insight. Riemann was so far ahead of his time that it was 30 years before anyone else began really to grasp his ideas——much less to have their own ideas of comparable value. In fact, Riemann was so far ahead of his time that the results which Siegel found in the private papers were a major contribution to the field when they were published in 1932, seventy years after Riemann discovered them. Any simplification, paraphrasing, or reworking of Riemann's ideas runs a grave risk of missing an important idea, of obscuring a point of view which was a source of Riemann's insight, or of introducing new technicalities or side issues which are not of real concern. There is no mathematician since Riemann whom I would trust to revise his work.

????? The perceptive reader will of course have noted the paradox here of a secondary source denouncing secondary sources. I might seem to be saying, "Do not read this book." But he will also have seen the answer to the paradox. What I am saying is: Read the classics, not just Riemann, but all the major contributions to analytic number theory that I discuss in this book. The purpose of a secondary source is to make the primary sources accessible to you. If you can read and understand the primary sources without reading this book, more power to you. If you read this book without reading the primary sources you are like a man who carries a sack lunch to a banquet.