By Emil Grosswald (auth.), Marvin I. Knopp (eds.)
Read 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
13)] to rewrite the third term in f(a)). As always the fact that f(a) = f(a - l ) is clear. 31 For condition ( i l ) of Lemma 2 we take PN : jqj-2N-l, N => 2. (-lqI4N+4;jqI4). (Iqj4N+3;Jqj2). lqI(2N+l)2+l(-jqI2;jqI2) 2(-jqI4;lqj4)~ (lql;lq[2)~(jqI2;lq[4)~(lql;[ql2)~ + 0 as N-~+-. 20) is identically zero w i l l now follow directly from Lemma 2 once we show that the various apparent simple poles are in fact removable singularities. CASE I. e. a2 § q-2N-l (where without loss of generality we may assume N ~ 0).
After Entry 18, Ramanujan lists all of the Bernoulli numbers with index < 38. All of the values are correct and agree with the table found in [ l , p. 810]. The next result is contained in (16) of . E~TRY 19(i). For each positive integer n, 2(22n-l)B2n is an integer. PROOF. We give a somewhat easier proof than that in . By the von Staudt-Clausen theorem, the denominator of B2n is the product of all those primes p such that (p-l)I2n. theorem, pl(2P-l-l). Let p be such an odd prime. Then by Fermat's But since (p-l)I2n, we have pI(22n-l), which completes the proof.
6) is valid for y = O. 6) are equal. 7), we easily obtain after a short calculation. =r ~2n). " r This completes the proof. 4), we find that the l e f t side of (2,31 may be written as (-2 sinh2(D/2}}nr = n=O 1 _ I+2 sinh2(D/2) l r r cosh D Since the given difference equation in operator notation is (E+E-l)f : 2r f : r D, the desired result follows. PROOF3. of ( l . l ) , or Let f(x+h)+pf(x) = r where p ~ -I. 3). PROOF. As before, since an , 0 ~ n < -, is independent of r (x) = ex. r we shall let Using ( l .
Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980 by Emil Grosswald (auth.), Marvin I. Knopp (eds.)