New Results Concerning Power Series Expansions of the Riemann Xi Function and the Li/Keiper Constants

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Proc. R. Soc. A (2008) 464, 711–731 doi:10.1098/rspa.2007.0212 Published online 8 January 2008

New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants

BY MARK W. COFFEY* Department of Physics, Colorado School of Mines, Golden, CO 80401, USA

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real N constants flkgkZ1 that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. A new representation of lk is developed in terms of the Stieltjes constants gj and the subcomponent sums are discussed and analysed. Accompanying this decomposition, we ﬁnd a new representation of the constants hj entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about sZ1. We also demonstrate that the hj coefﬁcients are expressible in terms of the Bernoulli numbersP Kj and certain other constants. We determine new properties of hj and sj, where sj Z rr are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function. Keywords: binomial transform; Li/Keiper constants; Riemann zeta function; Riemann xi function; Laurent expansion; Riemann hypothesis

1. Introduction

The Riemann hypothesis (Riemann 1859) and its extensions are well recognized to be among the most important problems of mathematical physics and analytic number theory. The apparent pair and higher correlation properties of the zeta zeros have served to intensify the study of random matrices and other areas of mathematical physics (Bogomolny & Keating 1995, 1996). This paper is concerned with the Li equivalence (Keiper 1992;Li1997, 2004) of the Riemann hypothesis that is described below. In this paper, we develop a new representation of the Li (Keiper 1992) constants lk (Li 1997). It is an explicit expression in terms of the Stieltjes constants gk and is based upon the work of Matsuoka (1985a, 1986). Correspondingly, we ﬁnd a new representation of the constants hj entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about sZ1. The subcomponent sums of lk are discussed and analysed. Based upon estimates for theP Stieltjes constants, we determine new estimates on j hj j Kj and jsj j, where sj Z rr are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function. Further properties of the sums sj are derived, including summation results. As the Li constants are the binomial transform of the sequence sj, this transform plays a role in some of our presentation. *[email protected]

Received 10 September 2007 Accepted 28 November 2007 711 This journal is q 2008 The Royal Society Downloaded from http://rspa.royalsocietypublishing.org/ on November 21, 2014 712 M. W. Coffey

In addition, we advance other subjects. We show that the hj coefficients are expressible in terms of the Bernoulli numbers Bj and certain other constants. We also (see appendix A) obtain analytic results and other properties for Lehmer (1988) sums over the non-trivial zeros of the zeta function. Moreover, we describe possible methods for linking hj, sk and the Li/Keiper constants with the statistical properties of Brownian motion. Our new representation of lk is direct. It complements the arithmetic formula (Bombieri & Lagarias 1999; Coffey 2004, 2005a) obtainable from the Guinand– Weil explicit formula (Guinand 1948; Weil 1980) or by other means. The Riemann hypothesis is equivalent to the Li criterion governing the N sequence of real constants flkgkZ1 that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. This equivalence results from a necessary and sufficient condition that the logarithmic derivative of the function x[1/(1Kz)] be analytic in the unit disc, where x is the Riemann xi function. The Li equivalence (Li 1997) states that a necessary and sufficient condition for the non-trivial zeros Z N of the Riemann zeta function to lie on the critical line Re s 1/2 is that flkgkZ1 is non-negative for every integer k. It is possible to use our approach also in pursuit of confirmation of the extended and generalized Riemann hypotheses. The corresponding l constants have been defined for Dirichlet & Hecke L-functions and other zeta functions (Li 2004), and the same leading behaviour O( j ln j ) has been found (Coffey 2005a, 2007a). Our attention here is only with the classical zeta function. There has been significant interest in the last few years in the Li/Keiper constants (Keiper 1992;Li1997, 2004) and the accompanying criterion for the Riemann hypothesis. We mention such literature that relates to the concrete application of this criterion. The lengthy reference by Coffey (2005a) proposed such an application, extracted the leading order of the Li/Keiper constants, conjectured their subdominant behaviour and presented limited numerical evidence and many other analytic results, including families of sums over the classical zeta function. An early version of these results was communicated to Lagarias, who has incorporated similar analysis into a setting of automorphic L-functions (Lagarias 2004). In the area of numerical calculation, contributions have been made by Mas´lanka (2004a,b), which supplement the original analytic and numerical efforts of Keiper (1992). In addition, Smith (1998) proposed a power series development closely related to the Keiper & Li ideas, and we have just recently analysed his coefficients cn (Coffey 2005b). Among the conclusions, we find that exactly cnZS2(n)/n, where S2(n), a subdominant summatory contribution to the Li/Keiper constants, is defined and discussed in the following.

2. Preliminary relations

The function x is determined from z by the relation xðsÞZðs=2Þðs K1Þ K p s=2Gðs=2ÞzðsÞ, where G is the gamma function (Riemann 1859; Edwards 1974; Ivic´ 1985; Titchmarsh 1986; Karatsuba & Voronin 1992; Davenport 2000) N and satisﬁes the functional equation x(s)Zx(1Ks). The sequence flngnZ1 is

Proc. R. Soc. A (2008) Downloaded from http://rspa.royalsocietypublishing.org/ on November 21, 2014 Riemann xi function and Li/Keiper constants 713 deﬁned by n 1 d nK1 l Z ½s ln xðsÞ Z : ð2:1Þ n ðn K1Þ! dsn s 1

The lj values are connected to sums over the non-trivial zeros of z(s) by way of (Keiper 1992; Li 1997) X 1 n ln Z 1K 1K : ð2:2Þ r r In the representation (Bombieri & Lagarias 1999; Coffey 2004, 2005a) n l Z S ðnÞ CS ðnÞK ðg Cln p C2ln2Þ C1; ð2:3Þ n 1 2 2 where g is the Euler constant and the sum Xn m n Km S1ðnÞ h ðK1Þ ð1K2 ÞzðmÞ; nR2; ð2:4Þ mZ2 m has been characterized (Coffey 2005a), n n 1 n n 1 ln n Cðg K1Þ C %S ðnÞ% ln n Cðg C1Þ K : ð2:5Þ 2 2 2 1 2 2 2

Further bounds on S1(n) have been developed by applying Euler–Maclaurin summations to all orders (Coffey 2005a). The summand of the quantity S1 can be written as a sum over the trivial zeros of z as N K X 1 X2 1 K Kn z K Z Z ð1 2 Þ ðnÞ 1 C n K n : ð2:6Þ Z ð2m 1Þ ð1 jÞ m 1 jZKN even The focus is then on the sum Xn n S2ðnÞ hK hmK1; ð2:7Þ mZ1 m where the constants hj can be written as ! K k XN kC1 ð 1Þ 1 k ln N hk Z lim LðmÞln mK ð2:8Þ N/N C k! mZ1 m k 1 and L is the von Mangoldt function (Riemann 1859; Edwards 1974; Ivic´ 1985; Titchmarsh 1986; Karatsuba & Voronin 1992; Ivic´ 1993), such that L(k)Zln p when k is a power of a prime p and L(k)Z0 otherwise. The constants hj enter the expansion around sZ1 of the logarithmic derivative of the zeta function

0 XN z ðsÞ ZK 1 K K p K ! K hpðs 1Þ ; js 1j 3 ð2:9Þ zðsÞ s 1 pZ0

Proc. R. Soc. A (2008) Downloaded from http://rspa.royalsocietypublishing.org/ on November 21, 2014 714 M. W. Coffey and the corresponding Dirichlet series valid for Re sO1is 0 XN z ðsÞ ZK LðnÞ s : ð2:10Þ zðsÞ nZ1 n

3. New representation of S2(n) and hj

In an earlier work (Coffey 2004), we showed low-order examples of how S2, and thus lk, could be written in terms of the Stieltjes constants gj. Here, we show how this can be generally done, based upon the work of Matsuoka (1985a, 1986). We then make some observations on the contributions of gj to these quantities. In the Laurent expansion of the zeta function about sZ1, XN K n Z 1 C ð 1Þ K n zðsÞ K gnðs 1Þ ð3:1Þ s 1 nZ0 n! and the Stieltjes constants gk (Stieltjes 1905; Hardy 1912; Kluyver 1927; Briggs 1955; Mitrovic´ 1962; Israilov 1979, 1981; Ivic´ 1985; Coffey 2006a) can be written in the form ! XN kC1 1 k ln N gk Z lim ln mK : ð3:2Þ N/N C mZ1 m k 1 If we put X Kk sk Z r ; ð3:3Þ r where the sum is over all the complex zeros of the zeta function, then N X s ln xðsÞ ZKln 2K ðK1Þk k ðs K1Þk ð3:4Þ kZ1 k and we see from equation (2.2) that the connection between the values ln and the s sequence { k}is ! Xn j n ln ZK ðK1Þ sj: ð3:5Þ jZ1 j It can be shown that (Zhang & Williams 1994; Coffey 2005a) k Kk sk Z ðK1Þ hkK1 Kð1K2 ÞzðkÞ C1; kR2 ð3:6Þ and ln p g s Z l ZK C C1Kln 2: ð3:7Þ 1 1 2 2

We have, from equation (3.5), ! Xn j n ln Z nl1 K ðK1Þ sj: ð3:8Þ jZ2 j

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Now, Matsuoka (1985a, 1986) has shown that Xj X Yh g Kj 1 jb sj Z 1Kð1K2 ÞzðjÞ Cj ; jR2: ð3:9Þ hZ1 h bZ1 jb! j1R0; .; jhR0 j1C/CjhZjKh Therefore, by combining this equation with equations (3.7) and (3.8) and using the deﬁnition (2.4), we obtain proposition 3.1. Proposition 3.1. n l Z 1K ðln p C2ln2KgÞ CS ðnÞ n 2 1 ! Xn n Xj 1 X Yh g K ðK1Þj j jb ; nR2: ð3:10Þ jZ2 j hZ1 h bZ1 jb! j1R0; .; jhR0 j1C/CjhZjKh Then, comparing with equations (2.3) and (2.7), we identify ! Xn n Xj X Yh g j 1 jb S2ðnÞ Z ngK ðK1Þ j ; nR2: ð3:11Þ jZ2 j hZ1 h bZ1 jb! j1R0; .; jhR0 j1C/CjhZjKh Similarly, by comparing equation (3.6) with (3.9), we obtain proposition 3.2. Proposition 3.2. Xk X Yh g k 1 jb hkK1 Z ðK1Þ k ; kR2: ð3:12Þ hZ1 h bZ1 jb! j1R0; .; jhR0 j1C/CjhZkKh Remark 3.3. (i) We note that if one were able to show that the right-most sum in equation (3.12) is always of one sign, then strict sign alternation of the {nj} sequence would follow. We prove this key fact alternatively in §4. (ii) As the value of h0 is Kg, the full summation term on the r.h.s. of equation (3.11) is equal to Xn n K hmK1 mZ2 m as per equation (2.7). As examples of equation (3.10), we have

2 2 l2 Z 2l1 K s2 Z 1 CgKg Cp =8K2ln2Kln pK2g1; ð3:13aÞ 1 3 2 l3 Z 3l1 K3s2 Cs3 Z 2 C p K6ln2K3lnpK12g1 2 4 7 Cg½3 C2ðgK3Þg C6g C3g K zð3Þ ð3:13bÞ 1 2 4

Proc. R. Soc. A (2008) Downloaded from http://rspa.royalsocietypublishing.org/ on November 21, 2014 716 M. W. Coffey and 2 4 2 l4 Z 4l1 K6s2 C4s3 K s4 Z 1 C2gK6g Kg K12g1 C12g1 K2g1 C6g2 2 3 p4 K2g K g C p2 C C4g2ðgK g Þ 2 3 3 4 96 1 7 K4ln2K2lnpK zð3Þ: 3:13c 2 ð Þ From equation (3.9), we see that there is always a term ngnK1=ðn K1Þ! present n in sn (coming from the hZ1 term). In addition, there is always a g term (coming from the hZn term). Therefore, in the expressions (3.10) or (3.11), there is always a contribution ! Xn n ðK1Þ j g j Z ð1KgÞn K1 Cgn: ð3:14Þ jZ2 j In particular, the term ng present in equation (3.11) is always cancelled by the corresponding term in this equation. We may note that already in Coffey (2004) j we had deduced that lj contains the term KðK1Þ ½ j=ðj K1Þ!gjK1. We see this term again arising from the jZn term in equation (3.10) for ln. It has been proved that (Berndt 1972) ½3 CðK1Þnðn K1Þ! jg j% ; nR1; ð3:15Þ n pn which has been improved to (Zhang & Williams 1994) ½3 CðK1Þnð2nÞ! jg j% ; nR1: ð3:16Þ n nnC1ð2pÞn n In addition, Matsuoka (1985a, 1986) gave an upper bound of O½ð1CgÞ on jsnj. This bound is not of direct use to us because it gives an exponentially large upper bound on ln. With the use of equation (3.15) and arguing as in Matsuoka (1985a, 1986), we are able to slightly improve the upper bound on jsnj and jhnK1j to O½ð1C1=pÞn=n. It may be of interest to interchange the two ﬁnite sums over j and h in equation (3.11), Xn Xj Xn Xn Xn Z C Z Z Z Z j 2 h 1 h 2 j h hZ1 jZ1 and see if useful information can be obtained in this way. We are leaving this investigation to future effort.

4. Further properties of the sequence sk and of hj

Here, we present various further properties of the sums of equation (3.3). First, the inverse relation to equation (3.5), expressing the sums sk in terms of the Li constants, is ! Xn k n sn ZK ðK1Þ lk: ð4:1Þ kZ1 k

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This equation follows from use of the! orthogonality! relation Xn k n k n ðK1Þ Z ðK1Þ dj;n; ð4:2Þ kZ1 k j where dp,q is the Kronecker delta. When inserting equation (2.3) into equation (4.1) in order to rederive equation (3.5), the following orthogonality property is useful: ! Xn k n k n ðK1Þ Z ðK1Þ dn;m: ð4:3Þ kZm k m The functional equation for the Riemann xi function generates summation relations for the sequence {sj}(Zhang & Williams 1994). When evaluated at sZ1 (or the symmetric point sZ0), the equation ln xðsÞZln xð1KsÞ and equation (3.4) give N X s k Z 0: ð4:4Þ kZ1 k A further example is the corresponding evaluation at sZ2 N N X s X s ðK1Þk k Z 2k k : ð4:5Þ kZ1 k kZ1 k Similarly, the logarithmic derivative of equation (3.4) gives XN s1 ZK sk ð4:6Þ kZ1 and XN XN k k ðK1Þ sk Z 2 skC1: ð4:7Þ kZ1 kZ0 Thus, the combination of equations (4.4) and (4.6) yields XN XN sk 1 l1 Z s1 ZK ZK sk: ð4:8Þ kZ2 k 2 kZ2 P N K Z In turn, equation (4.8) implies that kZ3ðð1=kÞ ð1=2ÞÞsk 0. As numerical examples, we have

s1 x0:023095708966121033814310247906495291621932127152051;

s2 xK0:046154317295804602757107990379077303530267962324145;

s3 xK0:00011115823145210592276266823891457847396418924898652;

s4 x0:000073627221261689518326771307030601511312831596274100 and K7 s5 x7:1509335576260773580109391313245132240664933152757!10 :

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From high precision values of the Stieltjes constants, one may also ﬁnd the hj coefﬁcients to high precision from a recursion relation (e.g. Coffey 2004; appendix). Then, equation (3.6) delivers the sums sk. The values of sk decrease quickly with k as described below, and equation (4.8) illustrates the relation l1 Zs1 ZKs2=2C/. Taking the j th derivative of x0ðsÞ=xðsÞ gives 0 ðjÞ XN x ðsÞ K K ZK ðK1Þks ðkK1ÞðkK2Þ/ðkKjÞðsK1Þk j 1 x k ðsÞ kZjC1 0 ðjÞ XN x ð1KsÞ K K ZK ZK ðkK1ÞðkK2Þ/ðkKjÞs sk j 1: ð4:9Þ x K k ð1 sÞ kZjC1

When evaluated at sZ1 this equation yields ! XN K jC1 k 1 sjC1 Z ðK1Þ sk: ð4:10Þ kZjC1 j When inserted into equation (3.5), we have yet another representation for the Li constants, which must be equivalent to equation (3.5) itself ! ! ! ! ! Xn n XN k K1 XN Xk n k K1 XN n Ck K1 ln ZK sk ZK sk ZK sk: jZ1 j kZj j K1 kZ1 jZ1 j j K1 kZ1 k ð4:11Þ

We next show how equation (4.10) directly gives a summatory relation for the coefﬁcients hj, demonstrating proposition 4.1. Proposition 4.1. For jR1 j j jC1 Kð jC1Þ ½1 CðK1Þ hj Z ðK1Þ Cf1K½1 CðK1Þ 2 gzð j C1Þ ! XN kC1 k C ðK1Þ hk: ð4:12Þ kZjC1 j We insert equation (3.6) into both sides of equation (4.10) and use the sum ! ! XN XN XN k K C k 1 ½1Kð1K2 ðk 1ÞÞzðk C1Þ ZK C kC1 kZj j kZj j mZ1 ð2m 1Þ XN ZK 1 ZK Kð jC1Þ C jC1 2 zð j 1Þ: ð4:13Þ mZ1 ð2mÞ For equation (4.13), we applied the summation (Coffey 2005a) ! N X k q j qk Z ; jqj!1: ð4:14Þ K jC1 kZj j ð1 qÞ

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Rearrangement of the terms gives ! XN j jC1 Kð jC1Þ kC1 k hj Z ðK1Þ Cf1K½1 CðK1Þ 2 gzð j C1Þ C ðK1Þ hk ð4:15Þ kZj j and then (4.12). In appendix A, we give an alternative method for deriving the summation results of this section for the sj values. We also establish connections with sums considered by Lehmer (1988). Matsuoka (1985b) has also shown the following:

2 K K s Z cosðn tan 1 2a Þ CO½ða2 C1=4Þ n=2; ð4:16Þ n 2 C n=2 1 2 ða1 1=4Þ where r1 Z1=2Cia1 is the first complex zero of z(s) and r2 Z1=2Cia2 is thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi second suchp complexffiffiffiffiffiffiffiffi pffiffiffi zero. For numerical purposes, we recall that 2 C x Z z 2= a1 1=4 2= 200 2=10 0:1414. Equation (4.16), together with equation (3.6) and the fact that for kR2

K ð1K2 kÞzðkÞ K1O1=3k; ð4:17Þ gives proposition 4.2. N Proposition 4.2. The sequence fhjgjZ0, with h0ZKg, has strict sign j alternation. That is, hj ZKðK1Þ cj, jZ0; 1; .; for some positive constants cj. In regard to equation (4.17), we have

XN K K 1 1 1K2 n z n K1 Z 2 nz n; 1=2 K1 Z C ; 4:18 ð Þ ð Þ ð Þ n C n ð Þ 3 jZ2 ð2j 1Þ where z(s, a) is the Hurwitz zeta function (cf. equation (2.6)). That is, equation (4.17) holds for all real kO1. The new result, proposition 4.2, was obtained by the author several years ago, as announced in Coffey (2005a).

5. Summation representations of the Stieltjes constants and {hj}

If we examine the relation (4.12) for j an odd integer, then the l.h.s. vanishes, while on the r.h.s., z( jC1) is expressible in terms of the Bernoulli number BjC1. We then suspect that the hj constants are expressible in terms of these numbers and other constants. In fact, we express here both the Stieltjes constants and the hk values in terms of Bernoulli numbers. Such relations are all the more important because the Bernoulli numbers have many known arithmetic and number theoretic properties.

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We have for integers kR0: Proposition 5.1. XN K n Xk kK[ Z 1 1 ð 1Þ kC1 K BkK[ C1 ln 2 gk C ln n k! K[ C [ ln 2 ðk 1Þ nZ2 n [Z1 ðk 1Þ! ! XN K nK1 ! ð 1Þ [ K BkC1 kC1 ln n C ln 2: ð5:1Þ nZ1 n k 1 We have for integers pR1: Proposition 5.2. h K K p 1 Z p!½L ðx ; .; x Þ CL ðy ; .; y Þ; ð5:2Þ p p 1 p p 1 p where Lp are logarithmic polynomials and XN K nK1 r r r! ð 1Þ r r xr Z ðr K1Þ!ðK1Þ Br ln 2 and yr Z ðK1Þ ln n: ð5:3Þ ln 2 nZ2 n The logarithmic polynomials are expressibleP in terms of partial Bell polynomials . Z n K kK1 K . Bn,k as (Comtet 1974) Lnðg1; g2; ; gnÞ kZ1 ð 1Þ ðk 1Þ!Bn;kðg1; g2; Þ. For the proofs of these two propositions, we make use of the alternating zeta function XN K nK1 Z K 1Ks K1 ð 1Þ s zðsÞ ð1 2 Þ s ; s 1; ð5:4Þ nZ1 n valid for Re sO0. We observe that XN K [ Ks Z 1 ð 1Þ [ K [ n [ ln n ðs 1Þ ð5:5Þ n [ Z0 ! and XN j K1 B C K 1Ks K1 Z 1 K ð Þ j 1 j K j K ! 2p ð1 2 Þ K C ln 2ðs 1Þ ; js 1j ; ð5:6Þ ln 2ðs 1Þ jZ0 ðj 1Þ! ln 2 with the latter expansion based upon the generating function for the Bernoulli numbers. We insert equations (5.5) and (5.6) into the r.h.s. of equation (5.4), and multiply and manipulate the infinite series. The first term on the r.h.s. of equation (5.6) provides the polar term of the zeta function, while the [Z0 term of a product of sums gives simply a factor of ln 2. Comparing to the defining expansion (3.1) for the Stieltjes constants yields equation (5.1). Remark 5.3. (i) It is probable that proposition 5.1 may be obtained by several other methods. (ii) For kZ0 in equation (5.1), aP rather classical expression for N n the Euler constant is obtained: gZð1=ln 2Þ nZ2 ðK1Þ ln n=nC ð1=2Þln 2. (iii) The convergence of equation (5.1) is such that more than 10 million terms over n are required to obtain the first few gj values to a handful of significant digits. A much faster converging, but more complicated, extension of equation (5.1) is given in proposition 6.1 of Coffey (2006b).

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For the proof of proposition 5.2, we insert equations (5.5) and (5.6) into the r.h.s. of equation (5.4) and take the logarithm "# N X ðK1ÞjB ln zðsÞ ZKlnðln 2ÞKlnðs K1Þ Cln 1 C j lnj 2ðs K1Þj jZ1 j

XN K nK1 XN [ C ð 1Þ K [ ln n K [ ln ð 1Þ [ ðs 1Þ : ð5:7Þ nZ1 n [ Z0 ! We then take into account the [Z0 term of the second line of equation (5.7) and compare with the integrated form of the defining expansion (2.9) for the hj coefficients, giving "# XN XN j h K ðK1Þ B K p 1 ðs K1Þp Z ln 1 C j lnj 2ðs K1Þj pZ1 p jZ1 j "# XN XN K nK1 [ C C 1 ð 1Þ K [ ln n K [ ln 1 ð 1Þ [ ðs 1Þ : ln 2 [ Z1 nZ2 n ! ð5:8Þ By making use of the defining expansion for the logarithmic polynomials (Comtet 1974, p. 140), equations (5.2) and (5.3) follow. Remark 5.4. Either directly through proposition 5.2 or by way of the Stieltjes constants, we see that the Li/Keiper constants of equation (2.3) are ultimately expressible in terms of the Bernoulli numbers and certain other constants such as powers of ln n where nR2.

6. A possible implication

The importance of having explicit representations for hkK1, sk, lk and S2(k), as in equations (3.9)–(3.12), rather than simply knowing that they exist, should not be underestimated. In this section we describe a possible path for the verification of the Riemann hypothesis based upon criterion (c) of Bombieri & Lagarias (1999). This criterion states that we need to show that there exists a constant c(e) such en that ln RKcðeÞe for every fixed positive e and each positive integer n in order for the Riemann hypothesis to hold. The essence of this criterion seems to be that we need to show that every Li/Keiper constant is bounded away from KN. Indeed, the Riemann hypothesis can fail under the Li criterion only if a lk becomes exponentially large and negative. In fulfilling this criterion, the crux of the matter is the magnitude of the hnK1 contribution to sn or S2. We describe a possible route for such estimation that depends heavily on known and needed estimates for the Stieltjes constants. On the r.h.s. of equation (3.12), the constrained sum over the indices j[ means that we have a partition of kKh over the non-negative integers. All such partitions are considered, meaning that their order does not matter. The number

Proc. R. Soc. A (2008) Downloaded from http://rspa.royalsocietypublishing.org/ on November 21, 2014 722 M. W. Coffey of such partitions is ! n K1 h K1 in h K . Then, using equation (3.15), we are left with considering sums of the form n 1 ! Xn 1 n K1 4h 1 n Z ½ð1 C4pÞn K1: ð6:1Þ K nKh pn hZ1 h h 1 p h This is an overestimation of hnK1, since the 4 factor may be reduced, jg0j=0!Zg!1, otherwise 1=jb %1 is conservative. In addition, the use of inequality (3.16) gives an improved estimation. For instance, equation (6.1) can be replaced with the expression ! Xk K h Yh 1 k 1 4 ð2jbÞ! k K C ; ð6:2Þ h K pk h jb 1 hZ1 h 1 bZ1 j b jb! where, by the duplication formula satisfied by the gamma function, K Z 1=2 jb C ð2jbÞ!=jb! p 4 Gðjb 1=2Þ. Thus,pffiffiffi for large k, we may expect an improvement over equation (6.1) by a factor of 2ð2=eÞk. Given proposition 4.2, verification of the Riemann hypothesis may be boiled down once again to estimation of the alternating binomial sum in equation (2.7). The more the values of hjK1 are away from uniformity, the less cancellation will occur in the sum S2(n). We recall (Coffey 2005a) that for large enough j, the j magnitudes jhjj cannot increase more rapidly than 1/3 .

7. A probabilistic setting for the s and h values

We have previously argued heuristically (Coffey 2005a) concerning the sum S2 of 1=2C3 equation (2.7) that S2ðnÞZOðn Þ for all 3O0. Indeed, very recent numerical 5 calculations (Smith 1998) for the first approximately 10 cnZS2(n)/n values appear to confirm this conjecture. Needless to mention, rigorous confirmation of this statement would provide verification of the Riemann hypothesis. In fact, this conjecture implying the Riemann hypothesis is stronger than it. In this section we provide yet another perspective. We describe probabilistic connections of the power series expansions employed in this paper. In particular, there are at least four, albeit related, points of view that could connect the h coefficients of equations (2.9) and (3.12), other series coefficients and the sum S2, with the theories of diffusion, Brownian motion and random walk. Underlying much of this connection is the fact that a Jacobi theta function (Coffey 2002), a solution of the heat equation, provides a basis for a (Mellin transform) representation of the Riemann xi function (Riemann 1859; Biane et al. 2001; Coffey 2005a, 2007c).

(i) The Stieltjes constants enter as cumulants (Abramowitz & Stegun 1964; Ehm 2001) of a probability distribution built upon Kszð1KsÞ. Indeed, we

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have by equations (2.9) and (3.12) (Matsuoka 1985a, 1986) that n Xn X Yh d K K ZK 1 gjb n ln½ szð1 sÞjsZ0 n! ds hZ1 h bZ1 jb! j1R0; .; jhR0 j1C/CjhZnKh

n ZKðK1Þ ðn K1Þ!hnK1; nR2: ð7:1Þ We recall that the cumulants have direct statistical meaning. As examples, the ﬁrst cumulant gives the mean of a distribution, the second cumulant the variance, the third cumulant the third moment or skewness and the fourth cumulant is related to the kurtosis. (ii) In the elementary relation (Biane et al. 2001; Coffey 2004, 2005a) ! XnK1 n K1 Z knKi ln n K ; ð7:2aÞ iZ0 i ðn iÞ!

with dn k h ln½xðsÞj Z ; ð7:2bÞ n dsn s 1

kn is the nth cumulant of the random variable L hlnð1=Y Þ, where EðY sÞZ2xðsÞ (Biane et al. 2001). Again, the cumulants are related to n n ðnÞ moments, mn ZEðL ÞZðK1Þ 2x ð0Þ. We also identify n kn ZKðK1Þ ðn K1Þ!sn: ð7:3Þ Substitution of this relation into equation (7.2a) returns equation (3.5). In turn, equation (3.6) directly links hnK1 with kn. EquationP (7.3) is evident 1Ks s N n by either comparing ln EðY ÞZlnðY ÞZln½2xðsÞZ nZ1 knðs K1Þ =n! with equation (3.4) orP by operating on the Hadamard product for the xi ðnÞ n n function, ½ln xðsÞ Z rðd =ds Þlnð1Ks=rÞ. Equations (7.2a), (7.2b) and (7.3) provide a tight framework for linking the non-negative random variable Y to properties of Brownian excursions (Biane et al. 2001). In this approach, it appears preferable to use supplementary relations such as equation (3.6), as equation (7.3) by itself displays none of the substructure of the Li constants. This may provide an avenue for relating the h values and S2 to, for instance, the asymptotic behaviour of random walks. (iii) A necessary and sufficientÐ condition for the Riemann hypothesis to hold is 2 Z that the integral ResZ1=2lnjzðsÞjjdsj=jsj 0, and this formula may be interpreted in terms of Brownian motion also (Balazard et al. 1999). Specifically, for two-dimensional Brownian motion starting from 0 (or Z C from 1), one may put ZT1=2 1=2 iYT1=2 as the first point of impact upon the critical line. Then, the vanishing of the integral just cited is equivalent z Z to the zero expectation value (Balazard et al. 1999) E½lnj ðZT1=2 Þj 0. This interpretation may provide another clue that the application of the theory of Brownian motion, for instance concerning first passage time, could yield very useful information for S2(n) and therefore for the Li constants.

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(iv) The Riemann hypothesis, and accordingly the Li criterion, are equivalent to the positivity of a certain Weil inner product (e.g. theorem 3.1 of Bombieri (2003)). This can lead to the investigation of the properties of iterated kernels of the diffusion equation. If the Green function Gðx; y; t0Þ of a certain diffusion equation is positive for some t0O0, the non-negativity of a linear functional follows, and then so does the Riemann hypothesis. An even better approach may be a direct study of the iterates of related kernels of the diffusion equation. In this case, what is needed is a demonstration of positivity of the asymptotic form of an integral that may be interpreted as over Brownian motion on the real line (Bombieri 2003, §8).

8. Summary and brief discussion

Our formulae for ln and S2(n) such as (2.3), (3.8) and (3.11) are effective. One can accordingly algorithmically generate the values for the Li/Keiper constants and all associated sequences and sums. The efﬁcacy of our approach is further shown by the very recent work of Coffey (2007b), wherein it is shown that the subsum S2(n) may be further decomposed. The relationship between the {sk} and {ln} sequences is that of the binomial transform, just as S2(n) is essentially the binomial transform of the sequence j fðK1Þ hjK1g. We have now shown the hj sequence to be of strict sign alternation, verifying a conjecture of Coffey (2005a). Therefore, we can now write the sum S of equation (2.7) as 2 Xn m n S2ðnÞ Z ðK1Þ jhmK1j: ð8:1Þ mZ1 m

We may note that, with the convention hK1 h0, this sum is an nth-order difference of the sequence fjhmK1jg. The alternating hk values are very far from arbitrary. Indeed, the functional equation of the xi function enforces the summatory relation of equation (4.12). We may expect this equation to have further analytic applications and be a possible useful check for numerical calculations. We have considered Lehmer sums (Lehmer 1988) over the non-trivial zeros of the zeta function. The Lehmer sums contain the first Li/Keiper constant l1 as one of their terms. We have also shown the connection between the hj coefficients and the Lehmer sequence {bm}. We have given some reasons why the behaviour of the sum S2 may be tied up with the theory of one- or two-dimensional Brownian motion. Indeed, the properties of cumulants and other statistical arguments provide another course for generating positivity results. The importance of an explicit formula for ln or hj should not be overlooked. For instance, in principle, only improved estimation of the Stieltjes constants prevents the verification of the Riemann hypothesis by way of either the Li criterion (Li 1997) or criterion (c) of Bombieri & Lagarias (1999). Based upon the work of Matsuoka (1985a, 1986), we have been able to, yet again, derive an arithmetic formula for the Li/Keiper constants, without recourse to the Guinand–Weil explicit formula.

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Appendix A. Lehmer sums over the non-trivial zeros of z

Lehmer (1988) considered the numerical calculation of the reciprocal-power sums of equation (3.3). We give a number of apparently novel summation relations in connection with his presentation. We also relate the h values of equation (2.9) specifically to his sequence of coefficients {bn}. We begin with his conclusion, wherein he illustrated the numerical values X XN 1 Z xK K sr 0:04619142; ðA1aÞ r rðr 1Þ rZ2 X XN 1 Z K r xK C ð 1Þ sr 0:04597052 ðA1bÞ r rðr 1Þ rZ2 and X XN 1 Z rK2 ! K a sr ; ja j a1; ðA1cÞ r rðr aÞ rZ2 where a1 ZIm r1 is defined as below equation (4.16). In equations (A 1a)–(A 1c), the sums are over all the non-trivial zeros of the zeta function. In equation (A 1c), we have introduced a condition, apparently not supplied by Lehmer, which ensures absolute convergence of the sums with parameter a. In regard to equation (A 1a), we easily have X X 1 ZK 1K 1 ZK ZK K K 2s1 2l1: ðA2Þ r rðr 1Þ r r r 1 Together, equations (A 1a) and (A 2) are equivalent to equation (4.6) of the text. Similarly, we have X X 1 Z K 1 C s1 C ðA3aÞ r rðr 1Þ r r 1 and ! X X 1 ZK 1 C 1 ! K s1 K ; ja j a1: ðA3bÞ r rðr aÞ a r a r Equations (A 1c) and (A 3b) can be restated as XN X K rK1 Z 1 C s sr K s1: ðA4Þ rZ2 r s r This equation follows directly from the differentiation of the relation (Lehmer 1988) X X XN r K s C s ZK s ln 1 r : ðA5Þ r r r r rZ2 rr

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Indeed, differentiating equation (A 4) j times with respect to s gives the identity sjC1 ZsjC1 when evaluated at sZ0 and relation (4.10) when evaluated at sZ1. The following proposition provides further closed form results for Lehmer’s sums (A 3a) and (A 3b). For this, we introduce the Glaisher constant A, given by "# Xn n2 n 1 n2 ln A Z lim k ln kK C C ln n C : ðA6Þ n/N kZ1 2 2 12 4 We then have proposition A.1:

Proposition A.1. For jaj!a1, XN 0 0 K x x K ak 1s Z s K ð1KaÞ Z s C ðaÞ; ðA7aÞ k 1 x 1 x kZ2 N X x0 1 ðK1Þks Z s K ð2Þ Z 12 ln AK Klnð4pÞðA7bÞ k 1 x kZ2 2 and X 0 1 Z x Z 1 C C K C C ð2Þ ð3 g 2ln2 24 ln A ln pÞ; ðA7cÞ r r 1 x 2 where g is the Euler constant, and for p2C XN s2 Kps1 Z ðp Ck K1Þsk: ðA7dÞ kZ1 Proof. Differentiating equation (3.4) gives 0 XN x K ðsÞ Z s K ðK1Þks ðs K1Þk 1: ðA8Þ x 1 k kZ2 We then put sZ1Ka, yielding equation (A7a). At aZK1, we have the relations x(2)Zp/6 and p p p 1 x0ð2Þ Z Kg K ln p C z0ð2Þ: ðA9Þ 4 2 12 p We then use the functional equation of the zeta function to change from z0(2) to z0(K1), recalling that z(K1)ZK1/12, and use the well-known relation ln AZ1=12Kz0ðK1Þ, giving p g 2 1 x0ð2Þ Z 1 C C ln 2K8lnA C ln p : ðA10Þ 4 3 3 3

Using the explicit form of s1 in equation (3.7) and noting the relation (A 3a) gives equations (A7b) and (A7c). From the functional equation of x0/x and equation (3.4) and multiplying by s p, we have XN XN k p kK1 pCkK1 ðK1Þ sks ðs K1Þ Z sks : ðA11Þ kZ1 kZ1 Differentiating this equation with respect to s and putting sZ1 gives equation (A 7d ). &

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In turn,P equation (A 7d )atpZ2 gives, as per equation (3.13a), 2s1 K s2Z ZK N C l2 kZ1 ðk 1Þsk. The result (A 7d ) may be extended by the multiple differentiations of equation (A 11) and putting sZ0 and 1. It may be additionally extended by multiplying successive differentiated relations by powers of s and evaluating at sZ0 and 1. Many other summatory relations follow from the above. For example, from equation (A 1c), we have

X XN [! K[K Z s ðr K2Þðr K3Þ/ðr K[ K1Þar 2 ðA12Þ K [C1 r r rðr aÞ rZ2 and from equation (A 7a), XN 0 ð[ Þ K[ K [ C x K ðk K1ÞðkK2Þ/ðkK[ Þak 1s Z ðK1Þ 1 ð1KaÞ : ðA13Þ k x kZ2 In particular, we ﬁnd

N X x0 0 p2 5 36 1 K ðk K1ÞðK1Þks Z ð2Þ Z K K ½z0ð2Þ2 C z00ð2Þ; ðA14Þ k x 4 2 kZ2 24 4 p p where again the functional equation of z could be applied on the r.h.s. In the course of proposition A.1, we used the value z0(2). This leads to proposition A.2. Proposition A.2. We have

XN K LðkÞ Z C K 2 g lnð2pÞ 12 ln A; ðA15Þ kZ2 k where L is the von Mangoldt function. Proof. We have by the Dirichlet series (2.10)

XN ln k p2 z0ð2Þ z0ð2Þ ZK Z ½g Clnð2pÞ K12 ln A Z zð2Þ 2 z kZ2 k 6 ð2Þ

2 XN ZK p LðkÞ 2 ðA16Þ 6 kZ2 k and the proposition follows. & By summation by parts, we have the alternative expression XN C 0 Z lnðr 1ÞK ln r z ð2Þ Hr C ; ðA17Þ rZ1 r 1 r where Hn are the harmonic numbers.

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Corollary A.3. We have XN hj ZK1Klnð2pÞ C12 ln Ax0:147176657996065667 ðA18Þ jZ1 and XN j ½1GðK1Þ hj ZK1G1Kð1G1Þlnð2pÞHg C12 ln A: ðA19Þ jZ1 Proof. From proposition A.2 and equation (2.9) at sZ2, we have XN hj ZK1KgKlnð2pÞ C12 ln A: ðA20Þ jZ0

As h0ZKg, equation (A 18) follows. Equation (A 19) follows by using equation (2.9) at sZ0. & Corollary A.3 gives the component sums N X g h ZK Klnð2pÞ C6lnAxK0:6339580366574063 ðA21Þ p 2 pZ2 even and N X g h Z K1 C6lnAx0:7811346946534720: ðA22Þ p 2 pZ1 odd

Many further summatory relations for the hj coefﬁcients may be obtained by proceeding similarly to proposition A.1 or by using equation (3.6). Lehmer (1988) introduced the function 0 0 XN g ðsÞ Z z ðsÞ C 1 Z n Z K K bns ; b0 ln 2p 1: ðA23Þ gðsÞ zðsÞ s 1 nZ0 In view of equation (2.9) and proposition 4.2 of the text, we easily ﬁnd XN XN n p p n p bn ZKðK1Þ ðK1Þ hp Z ðK1Þ jhpjðA24Þ pZn n pZn n and XN n hp ZK bn: ðA25Þ nZp p Equations (A 24) and (A 25) are the examples of another binomial transfor- mation pair. In a sense, bn is a complement to the sum S2(n) of equation (2.7). We have the consistent special cases XN XN p b0 ZK ðK1Þ hp and bn Z g ZKh0: ðA26Þ pZ0 nZ0

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In regard to this equation, we have the well-known values z(0)ZK1/2, z0ð0ÞZKð1=2Þln 2p, andP perhaps the not so well-known value (from equation 0 N p (2.9)) z ð0Þ=zð0Þ K1ZK pZ0 ðK1Þ hp. In the light of Lehmer’s relation m KmK1 bm Z ðK1Þ 2 zðm C1ÞK smC1; ðA27Þ for mR1(Lehmer 1988) and equation (A 25), we have XN n n KnK1 hp ZK ½ðK1Þ 2 zðn C1ÞK snC1; pR1: ðA28Þ nZp p As another example, we have from equation (A 24), differentiating equation (2.9) and evaluating at sZ0, XN 2 p 2 p b1 Z ðK1Þ php ZKg C K2g1 K1: ðA29Þ pZ1 12 2 Since b1 ZKp =24K s2 from equation (A 27), we see once again that 2 2 s2 Zg Kp =8C2g1 C1. Although it does not appear to be stated in Lehmer (1988), the expansion (A 23) holds in the disc jsj!2, the radius of convergence is 2. Consistent with m this, we see from equation (A 27) that for large values of m, bm zð1=2ÞðK1=2Þ . In addition, from equation (4.16), we have proposition A.4.

Proposition A.4. The sequence {bn} has strict sign alternation.

Illustration of the sign alternation and large-m behaviour of bm is given K10 in table 4 of Lehmer (1988). For instance, b30 z4:6566!10 , as expected. m KmK1 KmK1 Higher order approximations for large m are bm zðK1Þ 2 ð1C2 Þ and x K m KmK1 C KmK1 C KmK1 C KmK1 x bm ð 1Þ 2 ½1 2 Oð3 Þ Oða1 Þ,wherea1 14:134725. In turn, with these approximations, we have from equation (A 24) hpz Kð1=3ÞðK1=3Þp Kð1=5ÞðK1=5Þp C/, which is the expected behaviour in this context. Such straightforward geometric dependence upon p arises when we neglect the complicating perturbations arising from the sj constants. In fact, if we ignore the smC1 term of equation (A 26) altogether, we can easily sum the resulting series ! XN n XN K p zK K n KnK1 C ZK ð 1Þ hp ð 1Þ 2 zðn 1Þ C p nZp p mZ1 ð2m 1Þ p ðK1Þ K ZK ½K2 Cð2K2 pÞzðp C1Þ: 2 ðA30Þ In obtaining equation (A 28), we substituted the Dirichlet series of the zeta function, interchanged the resulting double sums and used again the series expression for the zeta function. It is the phase information in the s values that leads to ﬁne structure in the h values. We could further discuss the other sequences introduced in Lehmer (1988). For instance, there is another binomial transform pair between the Stieltjes constants and Lehmer’s dn values. However, we shall not pursue this here, only noting the two special exact values d0Z1/2 and d1Z(1/2)ln 2pK1.

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