The Sum of the Harmonic Series Is Not Enough. = a m π 2 + b m log 2 m.

Transcription

1 The Su of the Haroic Series Is Not Eough I Proble 6-7 Ovidiu Furdui cojectured that for each iteger there are correspodig ratioal ubers a ad b such that S : log H H a π + b log Here log x is the atural logarith of x The reader was asked to prove or disprove this cojecture Editorial ote The proble as stated is apparetly quite hard The solutios below deterie S for all but that does ot settle the issue For exaple oe fids that S 5 7π 3 5log 5 8 log φ where φ + 5 which suggests that the cojecture is false To be coplete however a couterexaple clai eeds to iclude a proof that there do ot exist ratioal ubers a ad b such that log φ aπ + b log 5 Neither the solvers or the editors have such a proof Solutio by Ora Kouba Higher Istitute for Applied Scieces ad Techology Daascus Syria We will prove that for each I particular log H H + 4 π log log si jπ S π log S π 3 4 log 3 S π 6 log 4 S π 5 8 log log S π log 6 4 log 3 log This aswers copletely the questio of evaluatig the sus S E-ail: ora kouba@hiastedusy

2 For let us defie Q t +t + + t Sice tq t t we have tq t Q t t ad cosequetly for adt that is Q t Q t t tq t t t t t Q t t Q t Q t t +t + t + + t +t + t + + t t t j t l We coclude that Q t Q t t dt l Q t Q t dt t j dt + l t l dt logh + H But the fuctios t Q tt Q t are positive ad cotiuous o [ ] so Q t Q t t dt Q t Q t t dt that is Q t Q t log t dt Now log t logt+logq t so log H H S Fially S Q t log t dt Q t Q t log t dt Q t Q t Q t logq t dt [ log Q t ] S Q log + J with J t log t dt Q k t

3 ad the proble is reduced to evaluatig the itegral J Let ω deote the th root of uity: e πi/ TheQ t t ωj ad It follows that hece J 3 J Q t Q t t ω j log t ω j t log t ω j t dt dt + log t ω j t dt log t ω j t dt We will use the followig lea the proof of which is postpoed to the ed Lea Let ΩC \ [ + [ that is the set of coplex ubers with a cut alog the set of oegative real ubers For z i Ω we defie F z by F z The F satisfies the fuctioal equatio z Ω Fz+F π z 6 log t z t where Log is the pricipal brach of the logarith dt Log z Log z With the otatio of the lea we ca write 3 as follows: J F ω j +F ω j π 6 π 6 Log ω j Log ω j Log ω j

4 For j<we have ω j sijπ/e iπ/π/ ad cosequetly Log ω j log sijπ/ + iπ j Therefore ad hece J 4 J π 6 log si jπ π j j π 4 Clearly follows fro ad 4 log sijπ/ Proof of the lea Note first that both F ad z Log z are holoorphic i the coected regio Ω ad sice Ω is ivariat uder the holoorphic appig z /z we coclude that z Gz F z+f +Logz Log z z is holoorphic i Ω so to prove the lea we have to prove oly that Gx π /6for each egative real x Nowforx we have itegratig by parts [ F log t x dt t x t x ] log t + dt x x t x x log x ad cosequetly G x for every egative real x This proves that for soe costat c wehavegx c for all x i the iterval Now lettig x ted to ad otig that li x F x we coclude that c F log t t dt This cocludes the proof of the lea t dt t dt π 6

5 Solutio by the Editor Uiversity of Mephis Mephis TN I view of the asyptotic relatio H log + γ + O/ it is evidet that the series defiig S does ideed coverge Let log H H F z : z The F is aalytic i the uit disc ad F S By Abel s Liit Theore [ p 4] S li x F x It reais to fid a suitable represetatio of F ad calculate the required liit Start with the well-kow series 3 H + z+ Log z z < where Log z log z + i Arg z deotes the pricipal brach of the logarith z < The for Gz : H z z + H +/ + z + + Log z+li z where Li z z is the dilogarith By the Multisectio Forula [ pp 97 98] if fz c z the M [fz] : c z fω k z ω e πi/ k I view of the relatio it follows that H z M [G z] F z log Log z M [G z] + Gz z < Split F z ito cotiuous ad sigular ters accordig to their behavior ear z The cotributio to S of a cotiuous ter is siply its value at z For exaple the dilogarith ters i Gz adm [G z] together cotribute Li E-ail: ccrousse@ephisedu 3 Differetiate Log z toget H z ad the itegrate π 6

6 The Multisectio Forula gives M [ Log z ] k Log ω k z The k ter aely Log z is sigular ad the reaiig tersare cotiuous The cotributio to S fro the latter is k Log ω k It is ecessary to calculate the liit of the su of all sigular ters as x : L li x log log x+ log x log x } To this ed set x e u Thex asu ad Asseblig all cotributios we have S log L li u log + u log e u /} li u log + u log e u / + Oe u } li u log + u u +log + Oue u } li log + Oue u } log π + 6 k Log ω k To siplify the forula for S ote that the pricipal logarith of ω k is Log ω k log ω k + i Arg ω k log si kπ + iπ Hece k Log ω k Re k k k Log ω k log si kπ π log si kπ π k k k

7 Substitutio of this result ito yields S + 4 π log k log si kπ REFERENCES [] L V Ahlfors Coplex Aalysis McGraw-Hill New York 966 [] Z A Melzak Copaio to Cocrete Matheatics Joh Wiley & Sos New York 973 Also solved by Thoas A Dickes Exxo Mobil Upstrea Research Copay Housto TX ad Nguye Va Vih ad Ngo Phuoc Nguye Ngoc studets Belarusia Uiversity Misk Belarus Editorial ote The approach of Thoas A Dickes was siilar to that i Solutio I additio to foral aalysis Dickes used a iteger-relatio detectio algorith to study uerically the possibility of represetig S by a liear cobiatio of π ad log with ratioal coefficiets Usig a itegral represetatio of S he applied the PSLQ algorith to look for itegers a b c such that as + bπ + c log Usig 5-digit arithetic this ethod gave the kow results for 3 ad 4 ad foud o solutios for 5 Nguye Va Vih ad Ngo Phuoc Nguye Ngoc first show that π S 6 k t k/ log t t t dt ad the use coputer algebra to evaluate the itegral for specific cases