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DEDEKIND SUMS AND LAMBERT SERIES L. CARLITZ 1. Introduction. Apostol [l] has proved a transformation formula for the function oo GP(x) = E »-p*m" (| x\ < 1), m,n=l where p is a fixed odd integer (1.1) c.(h, k) = E M(mod t) >1. In the formula occur the numbers 3P+1_,(4) £ (t) \ k / (P&s£p+1), \ k/ where (A, k) = l, the summation is over a complete (mod k), and Ss(x) is the Bernoulli function. Put (1.2) /(A, A:r) = E( )(*r- «=o \ then Apostol's s residue system A)»-c.(ft, *); / formula can be put in the form [2, §2] i2Tci)P (1.3) Gp(e*"0 = (*t - h)p-*GPie2™')+ „/ ,',/(*, *: 0, 2(/> + 1)! where r = (AV+*')/(>--A), that hh'+kk' (1.3) implies the following (1.4) /(A, ft; t) = r^/(- + l =0. It is also shown in [2] transformation formula for/(A, k; t) : *, A; - —) + — (5 + tB)"1. The purpose of this note is to give an elementary proof of (1.4) depending on the representation of eg(A, k) by means of "Eulerian" numbers (see (2.3) below). 2. It is convenient (2.1) to use, in place of (1.1), the fuller notation cr,,ih,k)= E MtIMt)' ,,(mod it) \ k/ \ k/ where now r, s are arbitrary non-negative integers; that c„(A, k) =cp+i^StSih, k). If we define the Eulerian thus it is clear numbers Hmia) by means of Presented to the Society, February 27, 1954; received by the editors January 1954. 580 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 19, DEDEKIND SUMS AND LAMBERT SERIES 1 - (2.2) a " - ex — a „ t, Xm = E Hm(a) — , m=0 then we have the representation (2.3) m\ [2, §6] BrB, rs cr..(h, k) =-E-■' ffr_i(f*) H^r1) where 5m denotes a Bernoulli number the summation is over all &th roots formula (2.3) is proved for r^l, 5^1 the right member as 0 for rs = 0, then valid for all non-negative r, s. In the " xry° P,,_o rlsl E k'CrAh, k)—^ = -" x in the even suffix notation and of unity distinct from 1. The but if we interpret the sum in it is easily verified that (2.3) is next place we have, using (2.2), y ~—,~ ez — 1 e»lk — 1 00 + *y£ n—^71—* x*i (1 - r*)(l ^ 581 y =-\gx _ 1 g»/* - %T 00 ySfe—n S #r(r*) - E ^.(r1) L— - f) r=o r! «-o v- f"1 f* 5! xy y.' ~ e* - f * eW* - f-1 1 and therefore oo (2.4) ~r ^.s yh >-1 E *'*..(*,*)- -y = *yE -^—- r,.=o r! where the summation 5! f .. e1 - f* e"'* - r1 > is now over all &th roots of unity. If we put (2.5) bT,s(h,*) = E (-!)-'( * ) *-«Cr+.-M(A, *) (-0 \ / / or, what is the same thing, (2• 6) cr,,(h, £) = E ( * ) A-'Jr+._M(A, *), (=0 W / then a straight-forward (2.7) " xr y* yields " (£x + hy)r E *'*..(*, *) — — = E »r..(*.*)-— r,«-o For brevity (2.8) computation r! 5.' T,.=0 we put fcs= kx+ hy License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use »"! y' —• 5! 582 L. CARLITZ [August and (2.9) Fih, ft; z, y) = E or..(A, k) — — ■ r,«-o rl si It follows from (2.4) and (2.7) that (2.10) Fih,k;kz,y) 3. If T)runs through (3.1) = xyE ^ f , ex - f* „„ evlhi — 1 ' the Ath roots of unity, then _^_r^-r_^. x* — 1 , x — ij , *ij — 1 Thus *f» = e- _ f* A = 1 e*r* _ ! j ^/^f-1 _ f - 1 "" , c1"1'?- f and (2.10) becomes xy ,-, (3.2) f 7?(A,k; kz, y) = — E- 1 But since f 1 / ex/*»?— f e"'*f — 1 the right member *? y, vex,h 1 \elMJ? — f \ 1 c1,/*f — 1/ exlh+«'kri — 1 of (3.2) is equal to <?exM_1_,^Zx^_ _J_L__ A 7r ex'hri - f ew»+w*, _i h Xi e»'*f - 1 ««/*+»/*,,- 1 Then using (2.8) and (3.1) we get kxy _ F(A, ft; kz, y) =- 1 E- i3.3) 1 kxy +- (C - l)(e' - 1) Returning to (2.14), we replace A, ft, x, y by ft, A, —ftx/A, 2 respectively; we get " (-ftx)r 2* ftxz -_ 1 1 (3.4) T,cr.(k,h)--=--EZU r! see http://www.ams.org/journal-terms-of-use s! A , «r*x'V* - 1 «''*ij - 1 License or copyright restrictions may apply to redistribution; 1954] DEDEKIND SUMS AND LAMBERT SERIES 583 But by (2.7) the left-hand member of (3.4) is equal to * (kz - kx)r E bT,(k, h)--—r,._o Comparison z* - r\ " (hy)T r,s=o r\ = E bF,(k, k)U^- s\ z' - • si with (3.3) leads at once to (3.5) zF(h, k; kz, y) = yF(k, h; hy, z) + (kz - hy) —-—- e' — 1 e" — 1 4. We now compare coefficients In view of (2.9) we get at once (4.1) of zTy" in both members rk'~1br-.1,,(k, k) = sh'-^-iAk, for all r, 5^0. Incidentally of (3.5). h) + rkBT-iBs - shBTBa-i this proves Theorem 1 of [2]. If we put (4.2) fm(h, k; r) = E (M ) (*t - *)—1-cM-...(A, k), —0 \ 5 / then it follows readily from (2.5) that (4.3) (kr - h)fm(h,k; r) - e(* V*r)—6Mi.(A, k). 8_0 \ 5 / In (4.3) replace h, k, r by —k, h, —1/r, respectively; then kr- h / 1\ —-—M- k<h'--) (4.4) ™ /w\/ A\m-' -£(.)(-t) »-■•<-*•*>• But it is clear from (2.1) that cr.,(- k, h) = (- \Ycr..(k, h) and therefore (2.5) implies (4.5) *,..(- 4, A) = (- \)%,,(k, h). Hence (4.4) becomes T^(kr(4.6) h)fj- k, h;-\ T = (-l)mE( )r»A—*_..(*, s=0 \ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5 / A). 584 L. CARLITZ It follows that (At - A)|/„(A, ft;r) - (- l)^rm-2fj= El ft, h; - i-U )rm_'^",_'&»'-».'(A' *)-*-h'-^i.^+iik, ,=o\ s / \ = E ( m ) rm- \ kBm-.B,_0 \ S / = ft E ( 1. ) kBrn-.+lBs-.X OT — 5+1 ) £—A - AE ( 8_0 \ 5 / h)\ m —s + 1 ,= 1 V ) ) t—5)M+15^i — 1/ = (k--.yB+TB)~ and therefore (4.7) /.(A, ft; r) = (- l)»r»-2/m (- ft, ft; - —) + — (5 + tB)'". In particular for m = p-\-l, p odd, (4.7) reduces to (1.4). We have therefore proved (1.4) and indeed the more general result (4.7) using only the representation (2.3) and familiar properties of the Bernoulli functions. References 1. T. M. Apostol, Generalized Dedekind sums and transformation formulae of cer- tain Lambert series, Duke Math. J. vol. 17 (1950) pp. 147-157. 2. L. Carlitz, Some theorems on generalized Dedekind Mathematics vol. 3 (1953) pp. 513-522. Duke University License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use sums, Pacific Journal of