By Emil Grosswald (auth.), Marvin I. Knopp (eds.)
Read Online or Download Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980 PDF
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Extra info for Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980
J. , 72 (1952), 474-500. 14. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomontl, fratrum Borntrager, 1829 (Reprinted: in vol. I, 49-239 of Jacobi's, Gesammelte Werke, Reimer, Berlin 1881-1891, now by Chelsea, New York, 1969). 15. N. , 13 (1974), 383-436. 16. J. Mordell, The value of the definite integral f " eat2+btdt/(eCt+d), Quarterly Journal of Math,, 68 C1920), 329-342. 17. J. Mordell, The definite integral f " eax2+bXdx/(eCX+d) and the analytic theory of numbers, Acta Math.
10)) is proved. 11). 11), we must find a suitable representation of this function. To effect this we consider the q-analog of Whipple's theorem [21; p. lO0, eq. e. g + =), e = -f = ql/2, d = c-l and a ~ I. O (q)n(-Cq)n (-c-I q)n (q;q2)~ (l+qn)(l+c-l)(l+c)q n(n+l)/2 (l+cqn)(l+c-lq n) n=l (I+ Z (q)~(-l)~ 1 (q;q2). 11)). Hence we shall replace Ramanujan's divergent series by the righthand side of the above expression. 30) is now routine. N ~ 2, for condition ( i i ) . One choose PN = lqI-N/2-1/4' The identity f(a) = f(a -I) is immediate.
593 [36, p. 589]. Entry I g ( i l i ) is a statement of the von Staudt-Clausen theorem, which we mentioned above and which is (lg) of . ENTRY 20. For each nonnegative integer n, (-l)n-IB2n+(-l)n(l-F2n) is an integer 12n, where F2n is the sum of the reciprocals of those primes p such that (p-l)I2n.