THE SECOND WEIGHTED MOMENT OF ζ. S. Bettin & J.B. Conrey

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1 THE SECOND WEIGHTED MOMENT OF ζ by S. Bettn & J.B. Conrey Abstract. We gve an explct formula for the second weghted moment of ζs) on the crtcal lne talored for fast computatons wth any desred accuracy. Résumé. Nous donnons une formule explcte pour le second moment pondéré de la foncton ζ sur la drote crtque permettant une évaluaton rapde pour n mporte quel degré de précson. For Iz > let let E 1 z) Introducton dn)enz) E 1 z) 1/z)E 1 1/z) : ψz). Then ψ s analytc n C, the complex plane mnus the negatve real-axs s gven n that regon by ψz) logz) γ ζs)ζ1 s) z s ds. z sn s Moreover, ψ satsfes a 3-term relaton 1/) ψz + 1) ψz) 1 z + 1 ψ z ) z Mathematcs Subject Classfcaton. 11M6. Key words phrases. Remann zeta functon, Moments. Research supported n part by a grant from the Natonal Scence Foundaton.

2 4 The second weghted moment of ζ for z 1 satsfes, for m 1, wth b 1 log1 + z) 1 ψ1 + z) z 1 + z a m 1 m b m + for n. The a m are extremely small: a m 5/4 3/4 e m m 3/4 m j b n ζn)b n n m 1 j a m 1) m z m ) b j+ sn m ) + O 1 m )) so that the seres a m z m converges every on ts crcle of convergence z 1. All of these results can be found n [BC]. Theorem 1. Let. The second moment of ζ Iδ) Then, for < Rδ) < Iδ we have C δ) log1 e δ ) + 1) sn δ a m 1 m+1 + b m + m j C 3 δ) ζ1/ + t) e δt Iδ) C δ) + C 1 δ) + C δ) + C 3 δ) + C 1 δ) 1 sn δ m 1 j C δ) sn δ/ e δ/ 1 e δ + eδ/ δ + / + γ log ); a m e mδ, ) b j+ wth b 1 b n ζn)bn n for n ; 1) n dn)e n cot δ ; ζ1/ + t) e t snh tδ + cosh tδ Publcatons mathématques de Besançon - 13

3 S. Bettn J. B. Conrey 43 If δ s real ths smplfes to We can use the fact that Iδ) 1 + log sn δ ) sn δ + δ) cos δ + log γ) sn δ + 1 sn δ a m cos mδ ζ1/ + t) e t snh tδ a m γ + 1 log that cos mδ 1 sn δ U m 1 cos δ mδ ) sn, U m s the mth Chebyshev polynomal, to rewrte ths further. Corollary 1. Suppose that < δ <. Then Iδ) γ log log sn δ ) sn δ + δ) cos δ + log γ) sn δ a m U m 1 cos δ mδ ) sn ζ1/ + t) e t snh tδ In ths verson the frst term above s the readly recognzed usual man term. Note that the ntegral n the rght-h sde of ths formula can be rewrtten n terms of the orgnal ntegral I: ζ1/ + t) e t snh tδ dt 1) n+1 In + δ) In δ)). Another way to put t s f we let then wδ, t) : e δt + e t snh δt cosh δ)t, ζ1/ + t) wδ, t) dt γ log log sn δ ) sn δ + δ) cos δ + log γ) sn δ + a m U m 1 cos δ mδ ) sn. Snce a m e m we have the followng estmate for a precse evaluaton of Iδ) as δ. Corollary. Gven δ > N 1 we can compute Iδ) to an accuracy of 1 N n tme tδ, N) N. Publcatons mathématques de Besançon - 13

4 44 The second weghted moment of ζ Ths s farly remarkable n that t doesn t depend on δ! By contrast f one consders the assocated ntegral 1 δ ζ1/ + t) dt then the tme to calculate wll depend on δ n a sgnfcant way, say δ θ for some θ >. Fnally, usng the fact that when δ s real, the magnary part of C 3 δ) nvolves the weght w δ, t) we deduce a formula for the dvsor sum n C δ) n terms of the coeffcents a m : Corollary 3. For < δ < sn δ dn) 1) n e n cot δ 1 a m + a m 1 ) snm 1 sn δ )δ) + δ 4 sn δ 1 + log cos δ ) cos δ. Ths can be rewrtten as a m + a m 1 ) snm 1 )δ) δ cos δ sn δ 1 + log cos δ ) ) cos δ dn) 1) n e n cot δ. If δ 1 1, then the frst term of the dvsor sum on the rght-h sde of the formula s , as the frst two terms on the rght-h sde are That means that the sum on the left-h sde s up to an error of around , whch s the second term of the dvsor sum. Snce the terms a m are around e m t would take more than 6 mllon terms of the seres n m each wth thouss of dgts of accuracy) to numercally check ths. 3. Proof of Theorem 1 Assume frst of all that δ s real wth < δ <. We have Now so that Iδ) 1 1/+ 1/ ζs)ζ1 s)e δs 1/) ds e δ/ 1/+ 1/ χ1 s) ) s Γs)e s/ + e s/ ) I 1 δ) e δ/ Iδ) I 1 δ) + I δ) 1/+ 1/ ) s Γs)e s/ ζs) e δs ds χ1 s)ζs) e δs ds. Publcatons mathématques de Besançon - 13

5 S. Bettn J. B. Conrey 45 Now I δ) e δ/ I δ) e δ/ I 3 δ) e δ/ 1/+ 1/ ) s Γs)e s/ ζs) e δs ds. I 1 δ) I δ) I 3 δ) 1/+ 1/ 1/ 1/ ) s Γs)e s/ ζs) e δs ds ) s Γs)e s/ ζs) e δs ds. Thus, Iδ) I δ) + I δ) I 3 δ). Note that I δ) I 3 δ) are analytc for δ < /. We rewrte I I 3 as ntegrals over t as I δ) e δ/ I 3 δ) e δ/ 1/+ 1/ e s/ ζs)ζ1 s) cos s/ eδs ds ζ1/ + t) e t/ e /4 e δt cos dt 1/ + t) 1/ 1/ 1/+ e δ/ 1/ ζs)ζ1 s) e s/ cos s/ eδs ds e 1 s)/ ζs)ζ1 s) cos 1 s)/ eδ1 s) ds ζ1/ + t) e t/ e /4 e δt cos 1/ t) Next we wrte e δt cosh δt + snh δt e δt cosh δt snh δt. Also, for real t, Thus, e /4 cos 1/ + t) + e /4 e t/ cos 1/ t) e /4 cos 1/ + t) e /4 et/ cos 1/ t). I δ) I 3 δ) Returnng to I we have ζ1/ + t) e t snh tδ + cosh tδ I δ) e δ/ Res s1 ) s Γs)e s/ ζs) e δs + Jδ) e δ/ δ + / + γ log ) + Jδ) Publcatons mathématques de Besançon - 13

6 46 The second weghted moment of ζ Jδ) e δ/ + ) s Γs)e s/ ζs) e δs ds; note that ths s a place we need the temporary) assumpton that δ s real to ensure convergence of the ntegral on the new path. Expng ζs) dn)n s nto ts Drchlet seres nterchangng the summaton ntegraton, we obtan Now ths s Thus, Jδ) e δ/ dn) 1 e δ/ + dn)e ne δ e δ/ 1 E 1 e δ )). E 1 e δ ) E 1 1 e δ ) 1 1 e δ 1 4 Jδ) e δ/ Altogether we now have Recall that e δ/ 1 eδ sn δ/ ζ1/ + t) e δt dt e δ/ 1 e δ 1 4 Γs)ne δ ) s ds ) e δ E 1 1 e δ + ψ1 e δ ) ) ) n dn)e 1 e δ + ψ1 e δ ). ) ) n dn)e 1 e δ e δ/ ψ1 e δ ) 1) n dn)e n cot δ e δ/ ψ1 e δ ). e δ/ 1 e δ + eδ/ δ + / + γ log ) e δ/ ψ1 e δ ) sn δ/ + 1) n dn)e n cot δ log z + 1) ψz) + z z ζ1/ + t) e t snh tδ + cosh tδ a m 1) m z 1) m Publcatons mathématques de Besançon - 13

7 S. Bettn J. B. Conrey 47 so that ψ1 e δ ) log1 e δ ) + 1) 1 e δ ) e δ/ ψ1 e δ ) log1 e δ ) + 1) sn δ + 1 e δ ) + 1 sn δ a m e mδ a m e mδ. The asserton of the theorem now follows for real δ. But both sdes are analytc n the regon < R < Iδ <. Therefore, by analytc contnuaton the dentty of the theorem holds n ths larger regon of the complex plane. References [BC] Bettn, Sro; Conrey, Bran, Perod functons cotangent sums. Algebra Number Theory 7 13), no. 1, août 13 S. Bettn, Centre de Recherches Mathematques Unverste de Montreal, P. O Box 618, CentreVlle Staton, Montreal, Quebec H3C 3J7 E-mal : bettn@crm.umontreal.ca J.B. Conrey, Amercan Insttute of Mathematcs, 36 Portage Ave, Palo Alto, CA 9436 USA School of Mathematcs, Unversty of Brstol, Brstol, BS8 1TW, Unted Kngdom E-mal : conrey@amath.org Publcatons mathématques de Besançon - 13