XX-RAY OF ZHANG’S ETA FUNCTION
JEFFREY STOPPLEA
BSTRACT . A study of the level curves Re ( η ( s )) = ( η ( s )) =
0, for η ( s ) = π − s /2 Γ ( s /2 ) ζ (cid:48) ( s ) gives a new classification of thezeros of ζ ( s ) and of ζ (cid:48) ( s ) . We assume the Riemann Hypothesisthroughout. Introduction.
By the Cauchy-Riemann equations, the level curvesfor the real and imaginary parts of a holomorphic function form or-thogonal trajectories, and plotting these is an interesting way to vi-sualize the function. In [1], J. Arias-de-Reyna used the terminology‘X-ray’ for the level curves Re ( ζ ( s )) = ( ζ ( s )) =
0. In [10],Zhang named the function η ( s ) = π − s /2 Γ ( s /2 ) ζ (cid:48) ( s ) .(Levinson [7] had the Lemma below, but did not name the function.Conrey [3] used the notation η , but for a family of functions with aparameter.) This function has an interesting property with respect tothe zeros of ζ ( s ) on the critical line: Lemma. ( [7, (1.6)] or [10, Lemma 1] ) Suppose t > . Then we have ζ ( + it ) = if and only if Re ( η ( + it )) = . The lemma makes the level curves for η ( s ) of interest. The resultspresented here are inspired by examination of the level curves, butthe proofs are independent of the actual graphics. The key ingre-dient is a version of Zhang’s Lemma 2 and Lemma 4 in [10] whichmakes explicit all the implied constants. This is done in the finalsection.Arias-de-Reyna used monochromatic graphics, with thick and thinlines for the two level curves Re ( ζ ( s )) = ( ζ ( s )) = Mathematics Subject Classification.
Key words and phrases.
Zeros of the Riemann zeta function, zeros of the deriva-tive of the Riemann zeta function. a r X i v : . [ m a t h . N T ] S e p JEFFREY STOPPLE - F IGURE
1. Level curves for η ( s ) , − ≤ σ ≤
8, 0 ≤ t ≤ ( η ( s )) = ( η ( s )) > ( η ( s )) = ( η ( s )) > ( η ( s )) = ( η ( s )) < ( η ( s )) = ( η ( s )) < Mathematica these colors are H UE [0], H UE [1/4], H UE [1/2], H UE [3/4]respectively.)Throughout we assume the Riemann Hypothesis. For shorthandwhen referring to ‘the zeros’ of ζ ( s ) we mean the nontrivial zerosonly. The Riemann zeros ρ = + i γ of ζ ( s ) occur where the greenand purple contours cross the critical line. The zeros ρ (cid:48) of ζ (cid:48) are vis-ible everywhere four colors come together (exclusive of the doublepole at s = Theorem.
With the usual indexing γ < γ < . . . of the imaginary partsof the zeros of ζ ( s ) , every odd indexed zero lies on a contour Im ( η ( s )) < .Every even indexed zero lies on a contour Im ( η ( s )) > . -RAY OF ZHANG’S ETA FUNCTION 3 - F IGURE
2. Level curves for η ( s ) , − ≤ σ ≤
8, 0 ≤ t ≤ Proof.
This follows from the Improved Zhang Lemma below, whichsays that as t increases, the argument of η ( + it ) decreases byexactly π between consecutive zeros. A Mathematica calculation of η ( + i γ ) determines the parity of all the zeros. (cid:3) Classification of zeros.
Zeros of ζ (cid:48) ( s ) . Type 0: We will say a zero ρ (cid:48) of ζ (cid:48) ( s ) is of TYPE neither of thelevel curves Re ( η ( s )) =
0, Im ( η ( s )) > ( η ( s )) =
0, Im ( η ( s )) < ρ (cid:48) cross the critical line σ = ρ (cid:48) of ζ (cid:48) ( s ) is of TYPE exactly one ofthe level curves Re ( η ( s )) =
0, Im ( η ( s )) > ( η ( s )) =
0, Im ( η ( s )) < ρ (cid:48) crosses the critical line σ = ρ (cid:48) of ζ (cid:48) ( s ) is of TYPE ( η ( s )) =
0, Im ( η ( s )) > ( η ( s )) =
0, Im ( η ( s )) < ρ (cid:48) both cross the critical line σ = JEFFREY STOPPLE (To be completely precise, ‘crosses the critical line’ above should re-ally be replaced with ‘crosses the critical line above t = s =
1, which crosses thecritical line below t = ζ ( s ) .The Lemmas do not apply in this region.) These zeros could be fur-ther classified according to what the other two contours are doing,but I don’t (yet) see the utility. Zeros of ζ ( s ) . Type 1: We will say a zero ρ = + i γ of ζ ( s ) is of TYPE ( η ( s )) = ρ (cid:48) which is of type 1.Type 2: We will say a zero ρ = + i γ of ζ ( s ) is of TYPE ( η ( s )) = ρ (cid:48) which is of type 2.Figure 2 is Figure 1 with the Im ( η ( s )) = ζ ( s ) (curve crosses the critical line) and ζ (cid:48) ( s ) (curves of different colors meet) and their types. When bothbranches form a loop to the left, it is type 2. When they loop to theright, it is type 0. If the two colors extend in opposite directions with-out looping, it is type 1. In Figure 2, the first four zeros of ζ (cid:48) ( s ) havetype 2; the next four alternate between types 1 and 2. The first zeroof type 0 occurs at height about 113, with another at height about132. At height about 161 we have two consecutive zeros of type 1,but from the way the graphics are imported into Latex one can nottell, looks like it might be a type 2 and type 0. The breaks in thecurves are an artifact of the Mathematica C ONTOUR P LOT command;they could be eliminated by setting the parameters to sample morepoints.
Theorem.
Every Riemann zero is of either type 1 or type 2. Thus we havea canonical mapping from the zeros of ζ ( s ) to those of ζ (cid:48) ( s ) , which is twoto one on the type 2 zeros, and one to one on the type 1 zeros. Zeros of ζ (cid:48) ( s ) of type 0 are not in the image. The Riemann zeros of type 2 are canonicallygrouped in pairs.Proof. All this is clear except the first statement, which says the con-tours which cross the critical line from the left must terminate insome zero of ζ (cid:48) ( s ) . The alternatives we must rule out is continua-tion of the contour on to the right, or looping back to the left.For the first possibility, note that the contour arg ( η ( s )) = π (resp.arg ( η ( s )) = − π ) does not exist in isolation; it is part of a continuumwhich deform smoothly as the argument is varied. But the argument -RAY OF ZHANG’S ETA FUNCTION 5 of η ( s ) is increasing (as one moves up vertically in the plane) forRe ( s ) >
3, but decreasing for Re ( s ) <
0. They can only cross overeach other where the argument of η ( s ) is undefined, at a zero ρ (cid:48) .The second possibility is ruled out by the Improved Zhang Lemma,which says that the argument of η ( s ) decreases monotonically as onemoves up the critical line. (cid:3) Asymptotics.
Let N ( T ) = (cid:93) { type 1 zeros ρ = + i γ | < γ < T } .NB: This is a nontraditional notation for the meaning of N ( T ) . Let N ( T ) = (cid:93) { pairs of type 2 zeros ρ − , ρ + | < γ + < T } .We have classically(1) N ( T ) + N ( T ) = T π log (cid:18) T π (cid:19) − T π + O ( log T ) .For j =
0, 1, 2, let N (cid:48) j ( T ) = (cid:93) (cid:8) zeros ρ (cid:48) = β (cid:48) + i γ (cid:48) of type j | < γ (cid:48) < T (cid:9) .NB: The (cid:48) here does not indicate a derivative with respect to T . TheTheorem above implies N ( T ) = N (cid:48) ( T ) and N ( T ) = N (cid:48) ( T ) . Thuswe have from [2]:(2) N (cid:48) ( T ) + N ( T ) + N ( T ) = T π log (cid:18) T π (cid:19) − T π + O ( log T ) .Subtracting (2) from (1) gives(3) N ( T ) − N (cid:48) ( T ) = T π log ( ) + O ( log T ) .Subtracting (1) from twice (2) gives(4) N ( T ) + N (cid:48) ( T ) = T π log (cid:18) T π (cid:19) − T π + O ( log T ) .In particular we have Theorem.
There are infinitely many type 2 zeros of ζ ( s ) , and thus also of ζ (cid:48) ( s ) . At least one of the other types of zeros of ζ (cid:48) ( s ) is infinite in number. JEFFREY STOPPLE
Open Problems. (1) Understand the asymptotics of N (cid:48) ( T ) , N ( T ) , and N ( T ) .(2) Find an algorithm to determine the type of a zero ρ (cid:48) .(3) Find an algorithm to determine the type of a zero ρ = + i γ .(4) Examine the numerics of the spacing of ( γ + − γ − ) log γ + forpaired type 2 zeros 1/2 + i γ − and 1/2 + i γ + .(5) Examine the numerics of ( β (cid:48) − ) log γ (cid:48) for zeros ρ (cid:48) , sepa-rated by type. See, for example, Figure 6.1 in [4].(6) The open half of the conjecture of Soundararajan [8]: Can oneshowlim ( β (cid:48) − ) log γ (cid:48) = ⇒ lim ( γ + − γ − ) log γ + = Improved Zhang Lemma.
Let F ( t ) def. = − Re η (cid:48) η (cid:18) + it (cid:19) Let log η ( s ) be any choice of the branch of the logarithm in an openset which contains the critical line but no zeros of ζ (cid:48) . By the Cauchy-Riemann equations, F ( t ) = − d arg ( η ( + it )) dt . Lemma.
For t > , F ( t ) > .Proof. In [10, (2.4), (2.5), (2.6)] Zhang writes ζ (cid:48)(cid:48) ζ (cid:48) ( s ) = − s − + A + ∑ β (cid:48) > (cid:18) s − ρ (cid:48) + ρ (cid:48) (cid:19) + Σ ,where Σ = ∞ ∑ n = (cid:18) s − ρ (cid:48) n + ρ (cid:48) n (cid:19) .Here − ( n + ) < ρ (cid:48) n < − n is the unique real zero on ζ (cid:48) ( s ) in the interval, while ρ (cid:48) = β (cid:48) + i γ (cid:48) arethe complex zeros. (And, in fact, since we are assuming RH we canwrite β (cid:48) > Mathematica gives that ζ (cid:48)(cid:48) ζ (cid:48) ( ) = log ( π ) + − C − C + π
12 log ( π ) , -RAY OF ZHANG’S ETA FUNCTION 7 where C denotes the Euler constant and C the first Stieltjes constant.Thus A = log ( π ) + − C − C + π
12 log ( π ) − h (cid:48) h ( s ) = ∞ ∑ n = (cid:18) n − s + n (cid:19) − s − log ( π ) + C σ = η (cid:48) η ( s ) = Re ∑ β (cid:48) > s − ρ (cid:48) + Re ∞ ∑ n = (cid:18) s − ρ (cid:48) n − s + n (cid:19) − C − C + π
12 log ( π ) − + log ( π ) − C − Re (cid:18) s + s − (cid:19) + Re ∑ β (cid:48) > ρ (cid:48) + ∞ ∑ n = (cid:18) ρ (cid:48) n + n (cid:19) .For s = + it , − Re (cid:18) s + s − (cid:19) = + t .From the data provided by [6], we estimateRe ∑ β (cid:48) > | γ (cid:48) | < ρ (cid:48) ≈ ρ (cid:48) < ρ (cid:48) at height t about log ( t /4 π ) /2 π , we estimateRe ∑ β (cid:48) > | γ (cid:48) | > ρ (cid:48) < · · (cid:90) ∞ log ( t /4 π ) π t dt = ∑ β (cid:48) > Re 1 ρ (cid:48) < Mathematica we easily compute ∑ n = (cid:18) ρ (cid:48) n + n (cid:19) ≈ JEFFREY STOPPLE while we estimate ∑ n > (cid:18) ρ (cid:48) n + n (cid:19) < ∞ ∑ n = n < ∞ ∑ n = (cid:18) ρ (cid:48) n + n (cid:19) < < − F ( t ) > − ∑ β (cid:48) > Re 11/2 + it − ρ (cid:48) + + o ( ) is positive for t (cid:29)
1. The term o ( ) is, more precisely,21 + t − Re ∞ ∑ n = (cid:18) s − ρ (cid:48) n − s + n (cid:19) .We claim that − Re ∞ ∑ n > t (cid:18) s − ρ (cid:48) n − s + n (cid:19) > ∑ n > t Re (cid:18) − ρ (cid:48) n − n ( s − ρ (cid:48) n )( s + n ) (cid:19) = ∑ n > t ( − ρ (cid:48) n − n ) Re (( s − ρ (cid:48) n )( s + n )) | s − ρ (cid:48) n | | s + n | .With − n − < ρ (cid:48) n < − n , and 2 n > t , every term is positive.Meanwhile − Re ∞ ∑ n < t (cid:18) s − ρ (cid:48) n − s + n (cid:19) = ∑ n < t ρ (cid:48) n − | s − ρ (cid:48) n | + + n | s + n | .From | s − ρ (cid:48) n | > | s + n | we deduce − + ρ (cid:48) n | s − ρ (cid:48) n | > − + ρ (cid:48) n | s + n | ,so this sum is bounded below by ∑ n < t ρ (cid:48) n + n | s + n | > − ∑ n < t ( n + ) + t .The sum has t /2 terms, each less than 1/ t so this sum is boundedbelow by − t . We conclude that F ( t ) > − ∑ β (cid:48) > Re 11/2 + it − ρ (cid:48) + + + t − t , -RAY OF ZHANG’S ETA FUNCTION 9 and for t >
7, 0.340479 + + t − t > (cid:3) By Stirling’s formula we have that for ρ = + i γ a zero of ζ ( s ) ,Zhang’s Lemma 3 is, more explicitly, F ( γ ) ∼
12 log (cid:16) γ π (cid:17) .Zhang’s Lemma 4 becomes Lemma.
For n ≥ , (cid:90) γ n + γ n F ( t ) dt = π .R EFERENCES [1] J. Arias-de-Reyna
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The number of zeros for ζ ( k ) ( s ) , J. London Math. Soc.(2) 2 (1970),pp. 577-580.[3] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are onthe critical line , J. Reine Angew. Math. 399 (1989), pp. 1-26.[4] E. Due ˜nez, D. Farmer, S. Froehlich, C.P. Hughes, F. Mezzadri, T. Phan,
Roots ofthe derivative of the Riemann-zeta function and of characteristic polynomials , Non-linearity 23 (2010), pp. 2599-2621.[5] F. Ge,
The distribution of zeros of ζ (cid:48) ( s ) and gaps between zeros of ζ ( s ) , Advancesin Math. 320 (2017), pp. 574-594[6] R. Farr, S. Pauli, personal communication.[7] N. Levinson, More than of zeros of Riemann’s zeta function are on σ = The horizontal distribution of zeros of ζ (cid:48) ( s ) , Duke. Math. J. 91(1998), pp. 33-59.[9] E. Titchmarsh, The Theory of the Riemann Zeta Function, Oxford UniversityPress, 2nd ed., 1986.[10] Y. Zhang, On the zeros of ζ (cid:48) ( s ) near the critical line , Duke J. Math. 110 (2001),pp. 555-572. E-mail address ::