Value Distribution of L-Functions with Rational Moving Targets

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Advanes in Pue Mathematis 3 3 79-73 Pulished Online Deeme 3 (http://wwwsipog/jounal/apm) http://dxdoiog/436/apm33998 Value Distiution of -Funtions wh ational Moving agets Matthew Cadwell Zhuan Ye *

1 Advanes in Pue Mathematis Pulished Online Deeme 3 ( Value Distiution of -Funtions wh ational Moving agets Matthew Cadwell Zhuan Ye * ntelligent Medial Ojets n Nothoo USA Depatment of Mathematial Sienes Nothen llinois Univesy DeKal USA madwell@e-imoom * ye@mathniuedu eeived August 6 3; evised Septeme 6 3; aepted Otoe 3 Copyight 3 Matthew Cadwell Zhuan Ye his is an open aess atile distiuted unde the Ceative Commons Attiution iense whih pems unested use distiution and epodution in any medium povided the oiginal wo is popely ed n aodane of the Ceative Commons Attiution iense all Copyights 3 ae eseved fo SCP and the owne of the intelletual popety Matthew Cadwell Zhuan Ye All Copyight 3 ae guaded y law and y SCP as a guadian ABSAC We pove some value-distiution esults fo a lass of -funtions wh ational moving tagets he lass ontains Seleg lass as well as the iemann-zeta funtion Keywods: Value Distiution; Moving aget; -Funtion; Seleg Class ntodution We define the lass to e the olletion of funtions s s a n n satisfying amanujan hypothesis n Analyti ontinuation and Funtional equation We also denote the degee of a funtion y d whih is a non-negative eal nume We efe the eade to Chapte six of [] fo a omplete definions Oviously the lass ontains the Seleg lass Also evey funtion in the lass is an -funtion and the iemann-zeta funtion is in the lass n this pape we pove a value-distiution theoem fo the lass wh ational moving tagets he theoem genealizes the value-distiution esults in Chapte seven of [] fom fixed tagets to moving tagets heoem Assume that and is a ational funtion wh lim ss et the oots of the equation ss e denoted y i hen () Fo any >max d O as () Fo suffiiently lage negative * Coesponding autho 4 π d O e < as Poof of () t is nown that if s n then a n O as ; s n whee is the index of the fist non-zeo tem of the sequene of an s n wh t Sine lim s s thee exists > suh that s s fo e s > t follows that < fo all eal pat of zeos of the funtion s s We set z Pz Qz whee the degees of PQ ae p q espetively; and define s ss hus thee is > suh that is analyti in the egion s > sine is a meomophi funtion in wh the only pole at s We apply tlewood s agument piniple [3] to in the etangle : t whee ae paametes satisfying >max > hus s ds π i d whee the given ahm is defined as in tlewood s agument piniple [3] o pove ou esult howeve we fist deompose ou auxiliay funtion y Open Aess

2 7 M CADWE Z YE s s Ps : Ps s fo p q P s Qs () s s : s s fo p > q s Whout loss of genealy we may assume that pq wheneve p q sine we an always we N s P s s Q s s hoie of the paametes whih define the etangle Howeve the modifiation will guaantee in the ase of that PQ exhi polynomial gowth whih is neessay fo ou poof n the ase of p > q aleady exhis polynomial gowth and no suh adjustment is neessay We now integate the ahm of to get N fo s due to ou s s P s d so fo p q ds s sd so fo p > q whee the O tems ae the integals of the maximum ontiution fom wing s as a sum of ahms By ou hoie of oth P and ae analyti in Hene Cauhy s heoem gives sdso fo p q sds () sd so fo p q o onnet this integal wh tlewood s agument piniple [3] we note that the definion of guaan- tees that π i d πi π i d (3) n light of () and eause the quanty given in (3) is imaginay-valued we get fo πi im i i ag i d i i ag dt ag ag i i i d i i dt O i t t i : 4 fo instane d d ag d ag i d j j O O (4) We now estimate Fo lage enough we have fo t (sine pq ) P Q P Q hen fo lage enough t we find in a simila fashion that Sine we have the same estimate fo we find that d t O d dto t O whee the final ound follows fom Jensen s inequaly Open Aess

3 M CADWE Z YE 7 t is nown [] that fo lim >max d Hene n a n dt O n O unifomly in >max d We next move to estimate Fo suffiiently lage posive eal nume we have so sine and P P q Futhemoe Sine we may tae lage enough so that we may we using a aylo seies expansion in the etangle Fo we have afte taing eal pats that an e P n n an an e P n n n n (5) We now oseve that fo suffiiently lage and some onstant M we have fo t P n n P i N and n n lim sup n n M fo suffiiently lage n light of these ounds and the definion of we have (6) whee the last equaly holds eause ould e suffiiently lage eplaing P y in the aove omputations we see anaously that O Finally we estimate 3 and 4 We show the omputation fo 3 explily and note that the ound fo 4 follows anaously We fist suppose that i has exatly N zeos fo hen thee ae at most N suintevals ounting fo multipliies in whih e i is of onsta nt sign hus i N ag π (7) t emains to estimate N o this end we define hen g z z i z i g i i e i i so that if then g Now let and >max and hoose lage enough so that > hen z i > > fo z < showing that no zeos o poles of z i ae loate d in z < hus oth z i and g z ae analyti in z < etting nˆ denote the nume of zeos of g z in z we have fo nˆ ˆ n d d d nˆ ˆ n By Jensen s fomula nˆ d π i and so d g g e lo g π π g i nˆ e d π g (8) e an an dt n n n n P n n an an n n n n n O n (6) Open Aess

4 7 M CADWE Z YE By (5) g fom a popety of is ounded Futhe is lea funtions that we have B as as ; s A t t t fo some posive asolute numes A B in any vetial stip of ounded width he same estimate must hold fo g z as well hus the integal in (8) is O implying that ˆ O Sine the inteval D n D follows that N nˆ O Wh this ound we integate (7) to dedue that d d 3 ag N π O As peviously noted we may ound 4 in the same way hus we attain the desied ounds fo j 4 and Consequently the fist pat of the theoem is poved y using (4) Poof of () As in the po of of the fist pat of the theoem we onlude that thee exists a eal nume fo whih the eal pats of all -values satisfy < ; and also thee exist B > fo eah ational funtion suh that no zeos of s s lie in the quate-plane < Bt > As efoe we define the etangle s : t whee ae paametes satisfying < B > max > max Poeeding as in the poof of the fist pat of the theoem we see that πi i d ag ag 4 j j i d i d O : O t dt fo whee is defined as in () n the equation aove we note that we have hosen to ompute sepaately ndeed this is the only estimate that we will need Fo the integals j j 34 and the ounds given as in the poof of the fist pat of the theoem still hold Fist integal is unhanged On the othe hand the integals 3 4 have hanged y ou hoie of ut as we have done as efoe we still have the desied ound sine the only equiement is that we onside in a vetial stip of fixed width whih we have in this ase We now ound Sine < B we have y the funtional equation in the defi nion of funtion s s s s s s s s s s s s s s aing ahms we get s s s s s s Sine fo t > we have unifomly in (9) s d iπ d Q t expd O 4 t whee ae two onstants t follows fo s as t that d d iπ d d s Qt exp O Qt 4 t Qt O d t Q O t t Open Aess

5 M CADWE Z YE 73 We now onside the last tem in (9) Sine and noting limsup d t t < we have fo any > and t t d fo suffiiently lage hen we see the quotient when O t t d is lage enough so that heefoe we find that deg d s O s t ntegating in light of these estimates we see dt d d t Q dt to 4 he fist integal is d Q and the e seond integal is O fo suffiiently lage and negative y the method used to deive (6) Hene 4 d Q O e Wh the estimates fo the j s we have poved the se ond pat of the theoem EFEENCES [] J Steuding Value Distiution of -Funtions Nume 877 in etue Notes in Mathematis Spinge 7 [] H S A Potte he Mean Values of Cetain Diihlet Seies Poeedings ondon Mathematial Soiety Vol 46 No 94 pp plms/s [3] E C hmash he heoy of Funtions nd Edion Oxfod 939 Open Aess

Numerical approximation to ζ(2n+1)

Numerical approximation to ζ(2n+1) Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/