AN ASYMPTOTIC FORM FOR THE STIELTJES CONSTANTS γ k (a) AND FOR A SUM S γ (n) APPEARING UNDER THE LI CRITERION

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1 MATHEMATICS OF COMPUTATION Volume 8, Number 76, October, Pages 97 7 S 5-578(97-X Article electronically published on May, AN ASYMPTOTIC FORM FOR THE STIELTJES CONSTANTS γ k (a AND FOR A SUM S γ (n APPEARING UNDER THE LI CRITERION CHARLES KNESSL AND MARK W COFFEY Abstract We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions We first determine the leading asymptotic behavior of the Stieltjes constants γ k (a These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function We then use asymptotic results for the Laguerre polynomials L α n to investigate a certain sum S γ (n involving the constants γ k ( that appears in application of the Li criterion for the Riemann hypothesis We confirm the sublinear growth of S γ (n+n, which is consistent with the validity of the Riemann hypothesis Introduction Let ζ(s be the Riemann zeta function, and the function ξ, satisfying ξ(s = ξ( s, be given by ξ(s =(s/(s Γ(s/π s/ ζ(s, 8, 9, 33] Within the critical strip < Re s<, the zeros of ζ and ξ coincide The Li criterion 3] states that the nonnegativity of the quantities d n ( λ n = (n! ds n sn ln ξ(s] s=, for each n isequivalenttotheriemannhypothesis(rhthatallnontrivial zeros of ζ have real part / The λ n s are connected to sums over the nontrivial zeros ρ l of ζ(s by way of 9, 3] ( λ n = n ] ( ρl, l and thus satisfy λ n = λ n Forn>, the series in ( is absolutely convergent, while for n = the sum should be taken over complex conjugate pairs of zeros of increasing imaginary part The Li/Keiper 9] constants may be written in the form 3,, 5, 6] (3 λ n = S (n+s (n+ n γ +ln(π], where S and S are certain alternating binomial sums, and γ is the Euler constant The sum S (n is relatively easy to estimate Its leading order is O(n log n, and this appears to be that of λ n itself Although it is not necessarily required to verify the Li Received by the editor June 8, and, in revised form, September 8, Mathematics Subject Classification Primary A6, 3E5, M6 Key words and phrases Li criterion, Riemann hypothesis, Stieltjes constants, Hurwitz zeta function, Riemann zeta function, Laurent expansion, asymptotic form 97 c American Mathematical Society

2 98 CHARLES KNESSL AND MARK W COFFEY criterion, a precise estimation of the sum S (n is desirable and remains open Any subexponential bound for S (n would suffice to verify the Riemann hypothesis under the Li criterion Recently, Coffey has further given the decomposition, ], ( S (n =S Λ (n n]+s γ (n+n], where, as the notation suggests, S Λ is a certain summation over the von Mangoldt function Λ We shall review at the end of section 3 that the sum S γ is a binomial sum over the classical Stieltjes constants γ k 3], n ( k ( n (5 S γ (n γ k = (k! k k= t L n (ln tdp (t In this equation, L α n is the Laguerre polynomial of degree n and parameter α 3] and P (t is the first periodized Bernoulli polynomial A key determination of this paper is to provide a detailed asymptotic analysis of S γ (n+n Previously Coffey estimated S γ (n +n as O(n /, ] by using a known asymptotic form of the Stieltjes constants and the leading order form of the Laguerre polynomial Here we give a self-contained study that presents detailed asymptotic expressions for S γ (n+n These in turn can be estimated to verify the sublinear growth in n As we show, S γ (n +n = O(n 3/ It appears that a yet closer estimation would yield a S γ (n+n = O(n /+ɛ result with ɛ We shall also give asymptotic results for γ n (a, the Stieltjes constants for the Hurwitz zeta function ζ(s, a Knowledge of γ n (a is useful for several areas, including in analytic number theory in the study of ζ(s, a and Dirichlet L-functions Our result for γ n (a now reduces to the γ n ( γ n special case in ] The Laguerre polynomials are pervasive in formulating the Li criterion 7] They provide certain test functions for a Weil inner product whose nonnegativity is equivalent to the Li criterion In addition, Laguerre polynomials are widespread in numerous application areas, including random matrix theory, quantum mechanics, and many others For example, two important problems of quantum mechanics, that are indeed essentially equivalent, the higher-dimensional harmonic oscillator and hydrogen atom, have wavefunctions with Laguerre polynomial factors Hence, detailed asymptotic forms for these polynomials are of interest There have been previous asymptotic analyses of Laguerre polynomials, including 3, 5, 7] We shall use a -term asymptotic result from ] for L α n(x forx>, and we also mention a corresponding result for L α n( z forz> in 3] The paper proceeds as follows In section we introduce the constants γ k (a, analyze them for k, and illustrate the main result numerically In section 3 we perform an asymptotic analysis of S γ (n +n with the help of various asymptotic results for L α n(x forx > Finally, in section we numerically compare the asymptotic expressions with known values of S γ (n+n An asymptotic form for the Stieltjes constants γ k (a The Hurwitz zeta function ζ(s, a may be analytically continued to the whole complex s-plane Its only singularity is a pole at s = with residue Correspondingly, there is the Laurent series for s =, ( ζ(s, a = s + n= ( n γ n (a (s n n!

3 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION 99 In this expansion, γ n (a are called the Stieltjes constants, 8, 7,,,, 5, 6, 3, 35, 36],andγ (a = ψ(a, where ψ =Γ /Γ is the digamma function For the coefficients corresponding to the Laurent expansion for the Riemann zeta function, one denotes γ n ( = γ n In this section, we present the leading asymptotic form of these constants for f(n n Throughout we write f(n g(n when the limit relation lim n g(n = holds We put ( C n (a γ n (a a logn a We have Theorem Let v = v(n be the unique solution of the equation (3 π expv tan v] =n cos v v, in the interval (,π/, withv π/ as n Letu = v tan v with u(n log n as n Then we have for n, C n (a B n e na cos(πacos(αn + β+sin(πa sin(αn + β] ( = B n e na cos(αn + β πa, where A = log(u + v u u + v, B = π u + v (u + + v ], / + α =tan ( v u v u + v, and ( (5 β =tan v ( u tan v u + Formula ( holds as long as we stay bounded away from zeros of the cosine factor We note that, in view of (3, the functions A, B, α, β depend weakly on n as log n and log log n The leading order is, A log log n and B π (log n The case of γ n ( has been recently investigated by us in ] The additional oscillation of the asymptotic form of γ n (a in the parameter a has been anticipated in Proposition 5 of 8] Although Theorem is predicated on n, we find that it provides a useful approximation for even small values of n As was true for the special case a =, the result ( captures the rapid exponential growth in magnitude with n It additionally contains the oscillatory behavior with respect to both n and a The form ( explicitly exhibits the /-antiperiodicity of C n (a for large n, ie, for n wehavec n (a C n (a +/ It appears that in (5 and several later equations in Sections - of 5] a factor of n! is missing

4 CHARLES KNESSL AND MARK W COFFEY Proof Here we indicate the proof of Theorem, concentrating on the extensions to the argument of ] We start with the integral representation (36], pp 53-5 for n, (6 C n (a = P (x a logn x x (n log xdx, where P (x =B (x x] = x x] / is the first periodic Bernoulli polynomial With the change of variable t =logx, wehave ( (7 C n (a = P (e t at n e t n t dt Since P has the Fourier representation ] (pp 85, sin(πlx (8 P (x =, πl we may write { (9 C n (a = Im Lπ L= L= ( } expπile t + n log t]e πila e t n t dt We shall show below that the series over L is absolutely and uniformly convergent, which justifies exchanging the order of summation and integration in going from (7 to (9 We shall show that for n the L = term in (9 dominates the others (see also ] As in ] we put ( h(t πile t + n log t, and the saddle points occur for h (t = Therefore, they satisfy ( te t = ni Lπ, and are asymptotically given by t log n log log n + γ + δ For integers M, we have γ = ( M + πi log(lπ andδ = log log n/ log n γ/log n = o( This gives t M =logn log log n log(lπ ( + M + ( log log n πi+ + o(], M=, ±, ±, log n We find that e h(t M as a function of M is maximized at M =, and as a function of L, atl = More precisely, we have the estimate log e h(t M =Reh(t M ] = n log log n n log n log log n ++log(lπ] n (3 log n + ( n n log n log log n +log(lπ] + O R log 3 n ( M + π where the O R rough error term may omit some factors of log log n From the right-hand side of (3 we see that the terms in (9 with L are roughly exponentially smaller than the first term In terms of M, (3 is largest at M =,,

5 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION but we can also easily show that the original contour (t, can be deformed to a steepest descent contour that passes only through the saddle t In Figure we plot the curves Reh (t] = and Imh (t] = in the (x, y plane, with L = and t = x + iy The intersection points of these curves are the saddle points, and the figure captures 3 saddles in the range y =Im(t π, 3π] (here we used n =, The steepest descent (SD curve through the saddle t = u + iv is given by Imh(t] = Imh(t ] so that ( ( n tan y +πe x cos y = nv ( x u + + n tan v v u The right side of (, for n, is approximately nπ/( log n so that the SD contour starts at the origin roughly at the slope y/x = π/( log n, traverses the saddle in a nearly horizontal direction (since h (t is to leading order real and negative and winds up at t = + iπ/ In Figure we sketch the SD contour when n =,, along with the steepest ascent (SA contour that is also a branch of (, and which orthogonally intersects the SD contour at the saddle t (here t 376+6i y 5 x Figure The real and imaginary parts of the saddle point equation h (t =areplottedinthe(x, y plane, with L = and t = x + iy Three saddle points are present in the range y =Im(t π, 3π] Here, n =, in ( The saddle point calculation can also be used to estimate the Lth term in (9 for n fixed and L, and this shows the rapid decay with L of the summand in (9, and the absolute and uniform convergences of the series

6 CHARLES KNESSL AND MARK W COFFEY Contours 3 x 6 8 Figure The steepest descent and ascent contours intersecting at the saddle point t are shown Here, n =, We have h (t =Lπie t n/t,sothath (t = n/t n/t We put A(n = Relog t /t ]andα(n = Imlog t /t ] and find ] (5 e h(t =exp n (log t t = e na(n+iα(n] We then have from (9, n e t ] C n (a π ena(n Im t + einα(n e πia, ] π it (6 = n ena(n Im t + einα(n e πia, where the saddle point relation ( at L = has been used Therefore, we have ( π it C n (a n ena(n Re t + einα(n sin(πa ( ] it (7 +Im t + einα(n cos(πa

7 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION 3 We now use, with the definitions (5, the relations ( ( it t Re t + einα(n = Im t + einα(n (8a and = π B sinnα(n+β(n], ( ( it t Im t + einα(n =Re t + einα(n (8b = π B cosnα(n+β(n] Then ( results from (7 Putting t = u + iv in the saddle point relation ( at L =, and eliminating u, results in the equation (3 solely for v Remarks The representation (6 may be readily verified by substitution in the defining relation ( Then we obtain (9 ζ(s, a = a s s + a s s P (x dx, Re s> (x + a s+ Some discussion on how to add higher order corrections to the saddle point method employed above is provided in ] In particular, only the L =termis still required, but the Taylor expansion of h(t should be extended, as well as a more detailed treatment of the factors e t (n/t in (7 We have noted the near anti-periodicity of C n (a for large n A simple explicit example of this is the special case C n (/ γ n (/ γ n = C n ( for n Our asymptotic result also has relevance for determing the asymptotic form of expansion coefficients of other important functions of analysis and analytic number theory We briefly describe applications to the Lerch zeta function Φ(z, s, a and to Dirichlet L functions For instance, we may consider the Lipshitz-Lerch transcendent e πinx ( L(x, s, a = (n + a s =Φ(eπix,s,a, n= for complex a different from a negative integer If we take here x real and nonintegral, the sum in ( converges for Re s> (For x an integer in (, we can reduce to the Hurwitz zeta function A case of interest is x =/ Then we obtain the alternating Hurwitz zeta function, ( ( L,s,a = n= ( n (n + a s = s ( ζ s, a ( ζ s, a + ] Therefore, expansion at s = can be made in terms of a combination of differences of Stieltjes constants γ k (a/ γ k (a +/] More specifically, we have, with the superscript denoting differentiation with respect to the second argument of L, ( ( j ( j ( L (j,s,a =( j s ( l log l j ( ζ (l s, a ( ζ (l s, a + ], l l=

8 CHARLES KNESSL AND MARK W COFFEY yielding (3 L (j (,,a = ( j j l= ( j ( a ( ] a + log l j ( γ l γ l l Dirichlet L-functions L(χ, s are known to be expressible as linear combinations of Hurwitz zeta functions We have for Dirichlet characters χ of modulus k, ( L(χ, s = n= χ(n n s = k s k m= ( χ(mζ s, m, Re s> k This equation holds for at least Re s> If χ is a nonprincipal character, then ( converges for Re s> Again, derivatives at s = may be obtained as combinations of Stieltjes constants Numerical results Formulas (-(5 are easily implemented for numerical computations In Figure 3 the exact values of C 5 (a from Mathematica V7 are plotted versus a for <a In Figure, values of C 5 (a obtained from ( are plotted versus a in the same range Already for the small value of n =5,Theorem gives a very good numerical approximation In Figure 5, the ratio R n of the exact values of C n (/3 to the values obtained from ( is plotted for n =, 5,,,5 Over this range of n, C n oscillates and its magnitude varies from less than to greater than 36 This range of values of C n (/3 illustrates the rapid growth of the Stieltjes constants with n The only points where R n deviates significantly from is near the zeros of the expression in (, as we would expect C a 5 Figure 3 Exact values of C 5 (a are plotted versus <a

9 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION 5 C a 5 Figure Values of C 5 (a obtained from ( are plotted versus a in the same range R n n Figure 5 The ratio of the exact values of C n (/3 to the values obtained from ( is plotted versus n =, 5, 5,,5 3 Asymptotics of S γ (n+n First, we present a result going beyond Theorem 85 of 3], which is given in ], and can be obtained using integral representations and the saddle point method We have

10 6 CHARLES KNESSL AND MARK W COFFEY Lemma For x> and α> we have L α n(x = ( n { α/ e x/ sin nx α π π x (nx / + π ] +cos nx α π + π ] } M nx (3 + O(n, where (3 M = M(x; α =(α + x x + α 6 This applies for n with a fixed x and <x< For n the sum in (5 behaves as S γ (n n and here we seek to estimate S γ (n+n, which we will show to be at most of the order O(n 3/, with significant oscillations We first express (5 in terms of Laguerre functions, and obtain the following Lemma We have, with N = n, (33 S γ (n+n = where (3 F (m; N + log(m+ log m F (m; N +, m= L N (ydy m + L N log(m +], and also (35 F (m; N += z n { } z z z ] (m + πi ( z z m z (m + z dz, z where the contour is a small counterclockwise loop about z = Proof By the definition of P, from (5 we have (36 S γ (n = t L n (log tdt m= m L n (log m Then using L n ( = n we obtain m+ (37 S γ (n+n = t L n (log tdt ] m + L n log(m +] m= m The change of variable y =logtthen yields (33 with F as given in (3 To obtain (35 we apply the contour integral representation (38 e x L α n (x = z n ( x exp dz, πi ( z α+ z and let x =logt in (37 Then integrating over t, using m+ (39 t z z dt = t z z t=m+ m z t=m, we obtain (35

11 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION 7 We proceed to evaluate the summand in (33 for N and/or m This will involve separately treating several ranges of m and N, and we first consider m with N = O( Lemma 3 Let log(m+ (3 F (m; N +;α L α N (ydy log m m + Lα Nlog(m +] Then for N = O( as m we have (3 F (m; N +;α = ( ] m Lα+ N +O (log m m Proof We let y =log(m +,y =logm and since (d/dxl α N+ (x = Lα+ N (x, we have y F (m; N +;α = L α N (xdx e y L α N(y y = L α N+ (y L α N+ (y e y L α N (y (3 = L α N+ (y +Δ L α N+ (y e y L α N (y, where Δ y y = /m +/(m +O(/m 3 Then expanding Δ for m we have F (m; N +;α = (Δ + e y L α N (y + Δ Lα+ N (y +O(Δ 3 L α+ N (y ( ] ( ] = m + O m 3 L α N (y + m + O m 3 L α+ N (y +O(Δ 3 L α+ N (y ( ] = m + O m 3 L α N (y +L α+ N (y ] ( ] = m + O m 3 L α+ N (y (33 = m Lα+ N (log m +O ( m ] Here, we used the recursion L α N (x+lα+ N (x =Lα+ N (x Lemma Let F (m; n be as defined in (3 and let β = (log m/n LetAidenote the Airy function (eg, 8] Then we have the following β-dependent asymptotic forms: (i For β> we have (3 F (m; n ( n+ m where β ( z n m z, n π z β β z (35 z = β + β β ] = ( β β (ii For β, weletβ =+n /3 α,withα = O( Then we have (36 F (m; n ( n+ ( /3 ( α Ai 6 m 3/ n /3

12 8 CHARLES KNESSL AND MARK W COFFEY (iii For <β<, we have (37 F (m; n β 5/ sinnf(β+g(β], m 3/ πn ( β / where (38 z + = β + i β β], f(β = (39 β β + arg β + i β( β], g(β =arg( z + 3 ] π + ( arg z+ + (3 z+( z + The values of arg are such that arg ( π, 5π, with the left endpoint corresponding to β and the right endpoint to β The value arg = π corresponds to β =+ (iv For m, n with ξ fixed, where ξ = ξ(m, n is defined by n (3 ξ = (m + log(m +, we have F (m; n log(m +] / {(cos ξ sin n log(m ++ π ] π ξ +(ξ sin ξcos n log(m ++ π ]} (3 (v For n with m = O(, we have n / (33 F (m; n cos n log(m ++ π ] π m + log(m +] 3/ Proof We merely sketch the main points Note that some of these results also follow from the uniform asymptotic results for the Laguerre function in 5] But, here we wish to give the expressions for F (m; n in the simplest possible forms From Lemma 3 with α = and the representation (38 we have F (m; N = ( ] m L N (log m +O m m πi ( z 3 e n log z e log m (3 z dz The latter expression is suitable for a steepest-descents expansion, and the saddle points occur for (35 n z log m (z =, or z +(β z + = This equation has real roots for β β >, or β> The root inside the unit circle is given by (35, so that z =(β β β/ The steepest descent method gives (36 F ( n+ m ( z 3 ( z n m z n z + βn ] / (z 3, leading to (3

13 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION 9 From (35 we have that the saddle point z asβ So for β we expand the integrand of (3 about z = We introduce the scaling z += n /3 w and β =n /3 α and determine that (37 F ( n n /3 ( w 3 exp 6 m 3/ πi L αw dw, where the path L extends from w = e iπ/3 to w = e iπ/3 By applying a standard contour integral representation 8], (pp 53 for the Airy function, we obtain (36 Next, we consider when <β< In this case we obtain the oscillations in F Now the two saddle points, (38 z ± = β ± i β β], are complex conjugates lying on the unit circle For <β<wehavere(z ± > and for <β<wehavere(z ± < The function h(z = n log z + βn/(z has (39 d h(z =n dz z + so that at z = z +, using relation (35 we have (33 β ] (z 3, n h (z + = z + + z+ (z + = ρeiφ + The steepest-descent method gives (33 F (m; n ( ] ρ Im e n log z + βn exp e iφ +/ e iπ/ m πn z + Here, (33 ρ = z + + z + = and also from (39, (333 φ + =arg We obtain from (33 β β, ( z+ + z+( z + (33 F (m; n β / m 3/ z + 3 sinnf(β+g(β], πn ( β / where z + = β,andf and g are given in (39 and (3 Therefore, (37 results We can easily verify that as β, (37 asymptotically matches to (36 as α Similarly, as β in (3, the result matches to (36 as α,in view of the asymptotic form (335 Ai(z π z / e 3 z 3/, z

14 CHARLES KNESSL AND MARK W COFFEY To obtain (33 in Lemma we simply use the leading term in (3 and note that for m = O(, (336 F (m; n =L n log m] L n log(m +] m + L n log(m +] log(m +] m + L n n / (m + log(m +] 3/ m + π sin n log(m + π ] Then using sin(x π/ = cos(x + π/ yields the results Now observe that the expression in (33 is not summable over m, astheamplitude decays only as m / for m Furthermore, as m, (33 does not match to (37 as β Indeed, as β, (36 behaves as (337 m 3/ πn β 5/ sin n β + π ] = n3/ π m 3/ (log m] sin n log m+ π ] 5/ This means that another scale must be found between n with m = O(, and n, m with β = (log m/n > We analyze this new scale below, and this will lead to the expression in (3 To establish (3 we first use Lemma to obtain a two-term asymptotic approximation to F (m; n Lemma 5 We have the following refined asymptotic approximation for the summand of (33 We adopt the notation (338 sin m = sin n log m + π/], cos m =cos n log m + π/] Then F (m; n F(m; n where ( m log m (339 F(m; n = n / π (log m / sin m + n log m ( m + log(m + n / π log(m +] / sin m+ + n log(m + n / + π m + log(m +] 3/ cos m+ + log m log (m + n log(m + ( log (m ] cos m 6 ] cos m+ 6 ] sin m+ This expression is O(m 3/ as m Form =the first term in F (involving sin m and cos m must be replaced by Proof We let Δ(m; n denote the -term asymptotic form of L n (log m coming from Lemma Using Lemma we arrive at (3 m Δ(m; n n / π (log m / ( log m sin m + n / (log m / log m 6 cos m ]

15 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION FromLemmawealsohave L n log(m +] (n / ( = m + π m + log(m +] 3/ sin n log(m + π ( +M(log(m +;cos n log(m + π ] { ( cos n log(m ++ π = n/ + O(n ] π m + log(m +] 3/ ( +sin n log(m ++ π L n log(m +] m + 3 n log(m + 6 +log(m + log (m + n / π m + ] } + Expanding the cosine factor, using n = n /( n+,thengives = cos n log(m ++π/] (3 log(m +] 3/ ( log (m n log(m + 6 The use of (3 and (3 gives Lemma 5 We observe that for m, sin n log(m ++π/] Δ(m; n Δ(m +;n = n / cos π m(log m 3/ m n / log m 3 ] (3 sin π m(log m 5/ m +O n (n 3/ +O m (m 3/ 6 Here the symbol O n (O m is used to denote the order of magnitude for n (m ; but the above is the same as the negative of (3 with m+ replaced by m therein Thus for m (339 behaves as the second difference Δ(m; n Δ(m +;n+ Δ(m +;n +O(m 3/ d which is of the same order as dm Δ(m; n, and this is, apart from some logarithmic factors, O(m 3/ Thus (339 is summable over m We next simplify (339 and obtain (3 in Lemma We write sin m =sin n log(m ++ π ] + δ (33 =sin m+ cos δ +cos m+ sin δ where, for m large, δ n log m n log(m + n n (3 m log m (m + log(m + ξ Thus we replace sin m in (339 by sin m+ cos ξ cos m+ sin ξ and note that, for m large, m (35 n / (log m ξ / log(m + Then retaining only the leading order terms in (339 leads to the approximation in (3 This applies for n, m with ξ = O( But in fact it remains valid for n with m = O(, and reduces to (33 in Lemma in this limit Note that for m = O( and n large, ξ and ξ sin ξ ξ, and this dominates cos ξ, which remains O( Thus (3 reduces to π / log(m+] / ξ cos m+ which is precisely (33 Also, for m O( n (3 asymptotically matches (37, ]

16 CHARLES KNESSL AND MARK W COFFEY since it agrees with (337 Now we have ξ sothatξ sin ξ = O(ξ 3 while cos ξ ξ / and thus (3 becomes ξ 3/ π log(m + sin n log(m ++ π ] which agrees with (337 for ξ (hence δ We have thus shown that (3 provides an asymptotically correct approximation to F (m; n forn,both for m = O( and for m = O( n Also, for m, ξ and (3 is O(ξ 3/ =O(m 3/ so the expression is summable over m Next, we return to our main goal, which is to approximate S γ (n+n in (33 The approximations to the summand F are given in Lemmas and 5 Expressions (3, (36, and (37 apply for β>sothatm = e βn is exponentially large in n But in this range the order in m of F changes from O(m 3/ (cf (37 to (roughly O(m (cf (3 and thus the summand is uniformly exponentially small in n when β > Hence only the terms in m that have m = O( or m = O( n contribute to the leading term of S γ (n+n and we have Theorem For n and away from the zeros of S γ (n+n we have S γ (n+n (36 m= { ξ (cos ξ sin n log(m ++ π ] π log(m +] / +(ξ sin ξcos n log(m ++ π ]} where ξ = n/(m + log(m +] An alternative asymptotic form is (37 S γ (n+n F(m; n, where F is given by (339, andifm =,thetermin(339 proportional to sin m and cos m must be replaced by In section we test the numerical accuracy of both (36 and (37 We can also obtain the simple upper bound below Theorem 3 For n sufficiently large, m= (38 S γ (n+n s π n 3/, where (39 s m= 8996 log(m +] 5/ (m + 3/

17 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION 3 Proof To establish (38 we use the elementary inequality A sin ϕ + B cos ϕ A + B Then denoting the summand in (36 as F (m; n wehave (35 π F log(m + ξ (cos ξ +(ξ sin ξ But (35 (cos ξ +(ξ sin ξ =( cos ξ ξ sin ξ+ξ ξ ξ ( x = x( cos xdx x dx = ξ so that π F / log(m +] / ξ 3/ = n 3/ (m + 3/ log(m +] 5/ and Theorem 3 follows However, the numerical studies in section suggest that the actual order of magnitude of S γ (n+n is smaller than O(n 3/, as Theorem 3 does not take into account the oscillations of the summands in Theorem The numerical studies suggest that (38 is valid also for moderate n, and we conjecture that (38 is true for all n 5 We conclude by establishing the first equality in (5 We have that for < Re s<, (35 lim N Therefore, we find that N (353 lim N m= z z N m= ] m s N s = ζ(s s z z ] (m + z m z (m + z ] = ( z ζ z Having interchanged summation and integration in (33 with (35, it follows that S γ (n+n = z n ( ] πi ( z z ζ dz z = ( n ] w w (35 πi C w w ζ(w dw, where C is a vertical contour to the right of Re(w = / By employing the Laurent expansion of ζ(w, we recover the defining binomial expansion in (5 for S γ (n: S γ (n+n = ( ] n w ( l γ l (w l dw πi C w l! l= (355 = πi = n + C n l= ( w w ( l (l! γ l n dw n ( n l ( l ( n γ l l! l + l=

18 CHARLES KNESSL AND MARK W COFFEY Numerical studies As we have discussed, S γ (n +n exhibits oscillations, taking on both negative and positive values as n increases This oscillation contains some local substructure, such as the pairs of nearby maxima and minima near n = 5, and 55 in Figure 6 Also plotted there are the corresponding results using the forms (36 and (37 At this level of graphical resolution, the refined result (37 is virtually indistinguishable from the exact values Known and reliable values of S γ (n +n may be obtained based upon the high precision values of γ k due to R Kreminski ] The first of these values of γ k are valid to 5 decimal digits Further values of γ k are less accurate, with γ 5 valid to 9 decimal digits, and γ valid to about 86 decimal digits These were computed using a variation of the published algorithm in ]; but instead of using Newton Cotes integration a slightly faster approach using numerical differentiation ideas was used Accordingly, R Smith has now calculated S γ (n+n for n up to 3] This result is shown in Figure 7, along with the corresponding values from (36 and (37 The numerical results of Smith, obtained with the aid of Mathematica R 7, were simply calculated to significant figures The expression (36 is usually a lower bound for the exact values of S γ (n+n, while the refined result (37 virtually coincides with the exact values, in particular, well capturing the local substructure in both curvature and magnitude Included in Figure 8 are values of S γ (n+n and the results of (36 and (37, as well as the upper bounding curve n / + and the lower bounding curve n / It does appear that the values of S γ (n+n are very close to O(n /, in place of the conservative upper bound of Theorem 3 The inequality in (38 of Theorem 3 begins to hold as soon as n 5 As this onset is dependent upon the value of s, we briefly comment on it From (39 we have ( s = log 5/ + 3/ m= log(m +] 5/ (m + 3/ < log 5/ + 3/ (m + 3/ m= = + ζ(3/ log 5/ 3/ Therefore, we have the simple lower and upper bounds ( log 5/ + 3/ log 5/ / log 5/ <s 3/ < + ζ(3/ log 5/ 3/, ie, 838 <s < 878

19 AN ASYMPTOTIC FORM FOR A SUM S γ (n UNDER THE LI CRITERION n Figure 6 Exact values of S γ (n+n, (, and (5 for n up to n Figure 7 Values of S γ (n+n, (, and (5 for n up to

20 6 CHARLES KNESSL AND MARK W COFFEY n Figure 8 Values of S γ (n+n, (, (5, and bounding curves varying as ± n / (in black, dashed for n up to 5 Acknowledgements We thank R Smith for supplying the known values of S γ (n+n to n = We acknowledge partial support from NSA grant H References ] M Abramowitz and I A Stegun, Handbook of Mathematical Functions, Washington, National Bureau of Standards (96 ] W E Briggs, Some constants associated with the Riemann zeta-function, Mich Math J 3, 7- (955 MR76858 (7:955c 3] E Bombieri and J C Lagarias, Complements to Li s criterion for the Riemann hypothesis, J Number Theory 77, 7-87 (999 MR75 (h:9 ] M W Coffey, Relations and positivity results for derivatives of the Riemann ξ function, J Comput Appl Math 66, ( MR96 (k:39 5] M W Coffey, Toward verification of the Riemann hypothesis: Application of the Li criterion, Math Physics, Analysis and Geometry 8, -55 (5 MR7767 (6g:76 6] M W Coffey, Polygamma theory, the Li/Keiper constants, and the Li criterion for the Riemann hypothesis, Rocky Mtn J Math, 8-86, ( 7] M W Coffey, The theta-laguerre calculus formulation of the Li/Keiper constants, J Approx Theory 6, (7 MR388 (8k:338 8] M W Coffey, New results on the Stieltjes constants: Asymptotic and exact evaluation, J Math Anal Appl 37, 63-6 (6 MR958 (7g:6 9] M W Coffey, Series representations for the Stieltjes constants, arxiv:95 (9 ] M W Coffey, The Stieltjes constants, their relation to the η j coefficients, and representation of the Hurwitz zeta function, arxiv/math-ph/7633v (7; Analysis 99, - (

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