Mathematics. — ‘Qn Lamsert’s series”. By Prof. J. C. Kiuyver. 
(Communicated in the meeting of Sept. 27, 1919). 
Transforming LAMBERT’s series 
n= n n= 
Be EE EN t (n) 2” 
n=! 1— zn ji 
SCHLÖMILCa *) deduced the asymptotic expansion 
1 
Car 108 logs 5 i pi (ee 
Blz i +4— 9-5 (loss) rn (te, est whe vas 
log = 
z& 
suitable to calculate the value of 4 (z) for real values of z approaching 
+1. Wieert °), slightly changing the formula, obtained a somewhat 
more general expansion, which can be utilised, when z, tending to + 1, 
takes complex values, and Lanpau’*) has simplified the proof of 
Wiarrt’s result. HANsEN*) has shown that the circle ,z) =1 is a 
natural limit of Z (ze) and in his lectures Lanpav established the same 
result in a simple and direct way. LANDAU's proof is given in a paper 
by Knopp ®), who in that paper, and also in his dissertation "), discussed 
series of the more general type 
n= PAL 
NN a by, TES 
Assuming the coefficients 6, to fulfill certain restricting conditions, 
he could establish several cases in which the continuation of the 
function AN (ze) beyond the circle of convergence is impossible. In 
the present paper I propose to deduce a new asymptotic expansion 
1) Ueber die Lambertsche Reihe. Zeitschr. f. Math. u. Phys., Bd. 6, 1861, p. 407. 
*) Sur la série de Lambert et son application a la théorie des nombres. Acta 
Math, XLI,-1918, p., 197. 
3) Ueber die Wigertsche asymptotische Funktionalgleichung für die Lambertsche 
Reihe. Archiv der Math. u. Phys., Ill. Reihe, XXVII, 1918, p. 141. 
4) Démonstration de |’impossibilité du prolongement analytique de la série de 
Lambert et des séries analogues. Kong. Danske Vidensk. Selskabs Forth. 1907, p. 3. 
5) Ueber Lambertsche Reihen. Journal f. d. reine u. ang. Math., Bd. 142, 
1913, p. 283. 
6) Grenzwerte von Reihen bei der Annäherung an die Konvergenzgrenze. 
Berlin, 1907. 
