Download Analytic Number Theory: Proceedings of a Conference Held at by Emil Grosswald (auth.), Marvin I. Knopp (eds.) PDF

By Emil Grosswald (auth.), Marvin I. Knopp (eds.)

ISBN-10: 3540111735

ISBN-13: 9783540111733

ISBN-10: 3540389539

ISBN-13: 9783540389538

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Additional info for Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980

Example text

1), n-l On(P-l) = Z Fk+l(n)(l-p)k. 4), we find that, for 1 < j < n, n-I aj = Z k=n-j k (n_j) Fk+l(n). j) - Fk+l(n) j=n-k n-l = k=n! O (n;j)Fr_j(n)=r n, where we have employed Entries 6(i} and 6(li). PROOFOF (ii). The proposed formula follows from the inversion of Ci) [33, pp. 43-44]. S(n,r), where the numbers S(n,r) are Stirling numbers of the second kind [ l , p. 824]. Because( i i i ) is such a familiar property of the Stifling numbers of the second kind, we shall omit the proof. G1 PROOFOF (iv).

3) RamanuJan's notation is unfortunate because Ak depends upon n. co n For Re s > l, the Riemann zeta-function ~(s) is defined by ~(s) = n=l in Ramanujan's notation, COROLLARY 6. (i) r ~(k) = Sk. Let 1 < r < n. Then Ak: r n, k=l (ii) (iii) and r-l Ar = kZ O= (-l) k ([) (r-k) n, Ar/n~ is the coefficient of xn in (eX-l) r, -s ; 60 (iv) ~ (-I)k+l kn({(k+l )-I ) k=l : (-l)n+(-l)n2-n'IonCl)+ zn (-l)k+IAk~(k+l), k=l where On(1) is determined in Entry 8. PROOFOF ( i ) . 1), n-l On(P-l) = Z Fk+l(n)(l-p)k.

5) (-l)n(n~)Z Z (2n+l)' n=O Set y = sinb(D/2) = ( ~ - I / ~ ) / 2 . (2 sinh(D/2) )2n+Ir : (Note that (vrE-I/~)nr 2D~ eDl2+e_b/2. ) calculation shows that /y2~l = CeO/2+e-D/2)/2. 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.

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Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980 by Emil Grosswald (auth.), Marvin I. Knopp (eds.)


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