Chowla s Problem on the Non-Vanishing of Certain Infinite Series and Related Questions

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1 Proc. Int. Conf. Number Theory and Dscrete Geometry No. 4, 2007, pp Chowla s Problem on the Non-Vanshng of Certan Infnte Seres and Related Questons N. Saradha School of Mathematcs, Tata Insttute of Fundamental Research, Hom Bhabha Road, Mumba e-mal: saradha@math.tfr.res.n In honour of Professor R. P. Bambah on hs 80th brthday. Chowla s problem In 969, Chowla rased the followng ueston: Suppose s a ratonal valued, perodc functon mod where > s a prme number. Then does the nfnte seres S = n. never vansh? Throughout the paper, we assume that f s a non-vanshng number theoretc functon. Much earler, n 949, Chowla [C] hmself showed that S 0 f f s an odd functon and prme wth also prme. 2 Around 970, Segel removed the restrcton that s prme n the result of Chowla. 2 In 973, Baker, Brch and Wrsng [BBW] solved Chowla s ueston completely. In fact, Chowla hmself solved the ueston around the same tme. The result of Baker, Brch and Wrsng s more general whch we state below. Theorem A [BBW]. Suppose s an algebrac valued, perodc functon mod. Then the nfnte sum S defned n. does not vansh f f satsfes the followng condtons. fr= 0 f < gcdr,< The cyclotomc polynomal s rreducble over Qf,...,f. The affrmatve answer to Chowla s ueston s mmedate snce s vacuous when s prme and follows when f s ratonal valued as t s well known that s 7

2 72 N. Saradha rreducble over Q. It was shown n [BBW] that condtons and are necessary. For example, let = p 2 where p s prme and let f be defned by n s = p s 2 ζs where ζs s the Remann Zeta functon. For p = 2, we get =0 wth perod 4. Ths shows that s necessary. Let χ,χ be uadratc characters mod 2 wth conductors 3 and 4, respectvely,.e., fn mod 3 χn = f n 2 mod 3 0 fn 0 mod 3 and fn mod 4 χ n = f n 3 mod 4 0 fn 0, 2 mod 4. Let f = 2χ 3χ. Then S = 0 snce L,χ= π 2 3,L,χ = π 3. Here Qf,...,f2 = Q 3 and 2 = X 2 + X 4 = X 2 3 X + X X +. Thus s also necessary. 2. Transcendental nfnte sums The sum whenever the seres converges S can be wrtten as a lnear form n logarthms of algebrac numbers wth algebrac coeffcents. More precsely, S = n = s= r= frξ rs log ξ s where ξ = e 2π/. See [Le] or [ASST] for a proof. By the famous result of Baker [Ba] on lnear forms n logarthms S s ether 0 or transcendental. Ths was observed n [ASST]. From ths t follows that L,χs transcendental for any non-prncpal character χ mod. The non-vanshng of any such L,χ s a basc result n Drchlet s famous theorem on prmes n arthmetc progresson. The

3 Chowla s Problem on the Non-Vanshng of Certan Infnte Seres 73 transcendence of L,χfor odd characters χ s also well known for long by the class number formula for uadratc felds. In [ASST] several other nfnte seres were shown to be transcendental. For nstance, seres lke 3n + 3n + 23n + 3, F n 2 n represent transcendental numbers. Here F n denotes the n-th Fbonacc number. In the same year when Theorem A appeared, Lehmer nvestgated seres n connecton wth extended Euler constants. He showed that n + 2n + 4n + = π 3 ; 6n +...6n + 6 = log 2 8 log 3 7π 3. Now we know that the above example represents a transcendental number, thanks to the theorem of Baker. Thus the ueston of non-vanshng of S becomes mportant. For a survey of these results and other related results, we refer to [ASST], [A], [AS] and [Tj]. 3. Values connected wth the Gamma functon Let us consder the seres n s where f satsfes the condtons of Theorem A. Ths can be wrtten as { } { } = fa = fa. n s n + a a= n=0 s a= s n + a/ n=0 s The last seres wthn {}s the well known Hurwtz-Zeta functon and we have Thus S = = n a= n + a/ = s s Ɣ a/ + terms nvolvng s. Ɣa/ = lt s a= fa Ɣ a/ Ɣa/, fa { } s Ɣ a/ + terms nvolvng s Ɣa/

4 74 N. Saradha snce a=fa= 0 s a necessary and suffcent condton for the convergence of S. By the consderatons n Secton 2, we get that a= fa Ɣ a/ Ɣa/ s transcendental. The above connecton of S wth values of Ɣ /Ɣwas ponted out by Professor M. Ram Murty durng the author s lecture n the conference at Chandgarh. Ths connecton has been notced by Lehmer [Le] n 973. He has also connected S wth some extended Euler constants. Ths vew pont was developed n great detal n [RS]. 4. Necessary and Suffcent condton for the sum S to vansh In 982, Okada [O] provded a crteron of all functons f for whch of Theorem A holds and S = 0. Ths crteron s a system of ϕ + ω homogeneous lnear euatons n f,...,fwth ratonal coeffcents where ϕ s the Euler totent functon and ωs the number of dstnct prme dvsors of. Ths crteron of Okada was used by Tjdeman [Tj] to show that a S 0 f f s completely multplcatve b S 0ff s multplcatve and fp k < p for every prme p. In 2003, Saradha and Tjdeman [ST] re-phrased the crteron of Okada so that t s more convenent for applcaton. We gve ths crteron below. For any nteger n and any prme p, let v p n denote the exact power of p dvdng n. Theorem B [ST]. Let be rreducble over Qf,...,f. Let M be the set of postve ntegers whch are composed of prme factors of and let εr, p = { vp + p v p r f v p r v p otherwse. Then S = 0 f and only f fam = 0 f or every a wth 0 <a<, gcda, = 4. m and m M n=0 f rεr, p = 0 for every prme dvsor p of 4.2 r= Applyng Theorem B, explct condtons were gven n [ST] under whch seres lke n an + b n + b n + b 2 or an + b n + s n + s 2 n + s 3 do not vansh. Here a,b are algebrac numbers, b,b 2,s,s 2 and s 3 are ntegers. n=0

5 Chowla s Problem on the Non-Vanshng of Certan Infnte Seres 75 In hs thess Tengely [Te] has shown that condton 4. n Theorem B can be wrtten as a fnte sum ϕ fr= 0 r under the addtonal hypothess r f m = whenever v p m = v p n for every prme p Erdős problem The followng ueston s one of the myrad problems posed by Erdős see [A]: Suppose f s a perodc number theoretc functon wth perod such that { ± f n 0mod = 5. 0 otherwse. Then does the seres S vansh? The condton f= 0 n Erdős problem s necessary as shown by Tengely recently. After a computer search he has shown that = 36 s the least nteger for whch S = 0 when for n 36 takes the followng seuence of values {,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,...} Okada [O] used hs crteron mentoned n Secton 4 to show that f f satsfes 5., then S 0 whenever 2ϕ. 5.2 Thus S 0 whenever ω 2. In [S], condton 5.2 was relaxed as 2ϕ h 5.3 where h = max r p α wth = p α...pα r r. Ths relaxed condton enabled us to cover more values of for whch S 0. For example, t was shown that S 0 whenever 54 and 525, 735, 945. The condton 5.3 was used to show that all fa s wth gcda, = take the same value. In fact, ths holds f 4.3 s assumed. Thus we show Theorem 5.. Suppose f s a perodc number theoretc functon wth perod such that 5. holds. Let = p α pα r r. Assume further that fa= fbwhenever gcda, = gcdb, =. 5.4

6 76 N. Saradha Then S 0 provded 2ϕ> h0 where h 0 = mn r p α. Proof. Suppose S = 0. By Theorem B, 4.2 s vald. By the convergence of S,wehave f h = 0. h= Thus h= gcdh,= f h = h= gcdh,> f h 5.5 By the assumpton 5.4 Let be such that p α L.H.S = ϕ. 5.6 = h 0. By 4.2, we get f h + 2 f h + +α f h p h p 2 h p α h + α + f h = p p α h Further f p, then p h f h = 0. From 5.5, 5.6 and 5.7, we get ϕ = h= gcdh,> p h p α h f h δ p 2 h f h α 2 f h α + p p α h f h where δ = { 0 fp otherwse.

7 Chowla s Problem on the Non-Vanshng of Certan Infnte Seres 77 Thus Hence ϕ ϕ + δ p p 2 p 3 +α 2 p α = ϕ p + δ p p 3 p 4 p α + α + p + p p α = ϕ p α + δ p p α p + = ϕ + δ p p p. + p α + p p α p α p 2ϕ δ +. p p p If δ = 0, we have 2 p < 2, ϕ a contradcton. Let δ =. Then α > and mplyng 2 2pα p p α p + pα p p p whch s mpossble. As a conseuence of Theorem 5., we get the followng corollary. Corollary 5.. Suppose f satsfes the condtons of Theorem 5.. Then S 0 for all <7325. Further suppose ω = 3. Then S 0 except possbly when s of the form 3 α 5 α 2 7 α 3 wth α 3,α 2 2,α 3 2or 3 α 5 α 2 α 3 wth α 4,α 2 3,α 3 2or 3 α 5 α 2 3 α 3 wth α 4,α 2 3,α 3 2.

8 78 N. Saradha Proof. We have ϕ We check that for <7325, = p pr. ϕ 2 h Thus by Theorem 5., S 0 for <7325. Suppose ω = 3. We may assume by Okada s result, that ϕ/ /2. Hence we fnd that p,p 2,p 3 {3, 5, 7, 3, 5,, 3, 5, 3}. Let p,p 2,p 3 = 3, 5, 7. Then from 5.8, we get h0 2 whch gves h 0. Thus S 0, whenever {3 5 α 2 7 α 3, 3 α 5 7 α 3, 3 α 5 α 2 7, α 2 7 α 3 }. When p,p 2,p 3 = 3, 5, or 3, 5, 3, we get h 0 33 and 65, respectvely. Ths yelds the possbltes for as mentoned n the corollary for whch S may vansh. It wll be desrable to obtan the mprovement of Theorem 5. wthout the assumpton 5.4. References [ASST] S. D. Adhkar, N. Saradha, T. N. Shorey and R. Tjdeman, Transcendental nfnte sums, Indag. Math. N.S [A] S. D. Adhkar, Transcendental Infnte sums and related uestons, Number Theory and dscrete mathematcs, Proceedngs of conference, Chandgarh, 2000 Hndustan Book Agency, 2002, [AS] S. D. Adhkar and N. Saradha, Arthmetc nature of sums of certan convergent seres, Resonance, 7, no [Ba] A. Baker, Transcendental Number Theory, Cambrdge Unversty Press 975 [BBW] A. Baker, B. J. Brch and E. A. Wrsng, On a problem of Chowla, J. Number Theory [C] S. Chowla, The Non exstence of Non trval Lnear Relatons between the Roots of a Certan Irreducble Euaton, J. Number Theory [Le] D. H. Lehmer, Euler constants for arthmetcal progressons, Acta Arth., [O] T. Okada, On certan nfnte sums for perodc arthmetcal functons, Acta Arth [RS] M. Ram Murty and N. Saradha, Transcendental values of the dgamma functon, J. Number Theory,

9 Chowla s Problem on the Non-Vanshng of Certan Infnte Seres 79 [ST] N. Saradha and R. Tjdeman, On the transcendence of nfnte sums of values of ratonal functons, J. London Math. Soc [S] N. Saradha, On non vanshng nfnte sums, The Remann Zeta functon and Related themes: Papers n honour of Professor K. Ramachandra, Ramanujan Mathematcal Socety, Lecture Notes Seres, pp [Te] S. Tengely, Effectve Methods for Dophantne Euatons, Thess, Leden, 2005 [Tj] R. Tjdeman, Some applcatons of dophantne approxmatons, Number Theory for the mllennum, Proceedngs of conference, Urbana, IL, 2000 Vol III A. K. Peters, Natck, MA, 2002,