שיר על השערת רימאן
Where are the zeroes of zeta of s? G.F.B. Riemann has made a good guess; They´re all on the critical line, saith he, And their density´s one over 2 p log t. This statement of Riemann´s has been like a trigger, And many good men, with vim and with vigour, Have attempted to find, with mathematical rigour, What happens to zeta as mod t gets bigger. The efforts of Landau and Bohr and Cramer, Littlewood, Hardy and Titchmarsh are there, In spite of their effort and skill and finesse, In locating the zeros there´s been little success. In 1914 G.H. Hardy did find, An infinite number do lay on the line, His theorem, however, won´t rule out the case, There might be a zero at some other place. Oh, where are the zeroes of zeta of s ? We must know exactly, we cannot just guess. In order to strengthen the prime number theorem, The integral´s contour must never go near ´em. Let P be the function p minus Li, The order of P is not known for x high, If square root of x times log x we could show, Then Riemann´s conjecture would surely be so.
Where are the zeroes of zeta of s? G.F.B. Riemann has made a good guess; They´re all on the critical line, saith he, And their density´s one over 2 p log t. This statement of Riemann´s has been like a trigger, And many good men, with vim and with vigour, Have attempted to find, with mathematical rigour, What happens to zeta as mod t gets bigger. The efforts of Landau and Bohr and Cramer, Littlewood, Hardy and Titchmarsh are there, In spite of their effort and skill and finesse, In locating the zeros there´s been little success. In 1914 G.H. Hardy did find, An infinite number do lay on the line, His theorem, however, won´t rule out the case, There might be a zero at some other place. Oh, where are the zeroes of zeta of s ? We must know exactly, we cannot just guess. In order to strengthen the prime number theorem, The integral´s contour must never go near ´em. Let P be the function p minus Li, The order of P is not known for x high, If square root of x times log x we could show, Then Riemann´s conjecture would surely be so.
העובדה שהפונקציה הזאת לא מוגדרת "הרבה", לא מונעת קיום פונקציה קדומה אלמנטרית (כלומר, אינטגרל שאפשר להציג אותו ע"י נוסחא פשוטה). אבל למיטב ידיעתי, אכן האינטגרל של לוג של סינוס אינו אלמנטרי.