A MODIFIED BERNOULLI NUMBER D. Zagier The classical ... - Up To

Proof of (IV). Using the relation (8) between Bernoulli numbers and zeta values,we findBn ∗ = 1 n − n 4 − ∑ ( ) n+r−1 (r−1)!2r −1 (2πi) r ζ(r)and hencewhere2≤r≤nr evenBn ∗ −U n = 1 n − n 4 + 1 2 ζ( 1) ∑ ∞+ b n,l , (12)2b n,l := (−1) n/2−1 πY n (4πl)− ∑2≤r≤nr evenl=1( ) n+r −1 (r−1)!2r−1 (2πil) r − 12 √ l .On the other hand, standard formulas for Bessel functions (cf. [1], 7.2(15) and7.3(16)) give(−1) n/2−1 πY n (x) = 2R ( K n (ix) )√ ∫ π/2 ∞e −ix (t/2+ix) n−1 2 +e ix (t/2−ix) n−1 2=Γ(n+ 1 2 ) (ix) n e −t t n−1 2 dt0for n even and positive, and hence, after some simple manipulations, the formulab n,l = 2Γ(1 2 )Γ(n+ 1 2 ) ∫ ∞where f n (x) is defined for x > 0 by0(t n−1 e −t (8πlf nt√ )) 1 t − dt (13)4 πlf n (x) = (1+ix)n−1 2 +(1−ix) n−1 22(ix) n− ∑2≤r≤nr even( n−1)2(ix) −r .n−rNote that f(x) = (2x) −1/2 + O(x −3/2 ) as x → ∞, so that b n,l = O(l −3/2 ), as wealready know. Clearly f n (x) extends to C 0 := C(−i∞,−i][i,i∞) as an evenholomorphic function, and the binomial theorem gives the Taylor expansionf n (x) = ∑ ( n−(−1) r 1)2x 2r (|x| < 1).n+2rr≥0The beta integral identity( n−1)2= 1 n+2r πΓ(n+ 1 2 )Γ(2r+ 1 2 )Γ(n+2r+1)6= 1 π∫ 10u 2r−1 2 (1−u)n− 1 2 du