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Notes Relating to Newton Series for the Riemann Zeta Function NOTES RELATING TO NEWTON SERIES FOR THE RIEMANN ZETA FUNCTION LINAS VEPSTAS ABSTRACT. This paper consists of the extended working notes and observations made during the development of a joint paper[?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omis- sions are my own fault. This set of notes contains considerably more content, although is looser and sloppier, and is an exploration of tangents, dead-ends, and ideas shooting off in uncertain directions. The finite differences or Newton series of certain expressions involving the Riemann zeta function are explored. These series may be given an asymptotic expansion by con- verting them to Norlund-Rice integrals and applying saddle-point integration techniques. Numerical evaluation is used to confirm the appropriateness of the asymptotic expansion. The results extend on previous results for such series, and a general form for Dirichlet L-functions is given. Curiously, because successive terms in the asymptotic expansion are exponentially small, these series lead to simple near-identities. A resemblance to similar near-identities arising in complex multiplication is noted. 1. INTRODUCTION The binomial coefficient is implicated in various fractal phenomena, and the Berry con- jecture [ref] suggests that the zeros of the Riemann zeta function correspond to the chaotic spectrum of an unknown quantization of some simple chaotic mechanical system. Thus, on vague and general principles, one might be lead to explore Newton series involving the Riemann zeta function. Of some curiosity is the series 1 X∞ b (1.1) ζ(s) = + (−1)n n (s) s − 1 n! n n=0 where (s)n = s(s − 1) ··· (s − n + 1) is the falling Pochhammer symbol. Note the general resemblance to a Taylor’s series, with the Pochhammer symbol taking the place of the monomial. Such general types of resemblances are the key idea underlying the so-called “umbral calculus”. Here, the constants bn play a role analogous to the Stieltjes constants of the Taylor expansion. The bn have several remarkable properties. Unlike the Stieltjes constants, they may be written as a finite sum over well-known constants: n µ ¶ 1 X n (1.2) b = n(1 − γ − H ) − + (−)k ζ(k) n n−1 2 k k=2 for n > 0. Here Xn 1 (1.3) H = n m m=1 Date: November 14, 2006. 1 englishNOTES RELATING TO NEWTON SERIES FOR THE RIEMANN ZETA FUNCTION 2 are the harmonic numbers. The first few values are 1 (1.4) b = 0 2 1 (1.5) b = −γ + 1 2 1 (1.6) b = −2γ − + ζ(2) 2 2 (1.7) b3 = −3γ − 2 − ζ(3) + 3ζ(2) 23 (1.8) b = −4γ − + ζ(4) − 4ζ(3) + 6ζ(2) 4 6 Although intermediate terms in the sum become exponentially large, the values themselves become exponentially small: ³ √ ´ 1/4 −2 πn bn = O n e The development of an explicit expression for the asymptotic behavior of bn for n → ∞ is the main focus of this essay. The principal result is that µ ¶ µ ¶ 2n 1/4 ³ √ ´ √ 5π ³ √ ´ (1.9) b = exp − 4πn cos 4πn − + O n−1/4e−2 πn n π 8 The method of attack used to obtain this result is to convert the Newton series into an integral of the Norlund-Rice form[?], which may be evaluated using steepest-descent / stationary-phase methods [need ref]. These techniques are sufficiently general that they can be applied to other similar sums that appear in the literature [multiple refs] (See penultimate section below, where these are reviewed in detail). One of the principal results of this essay is to apply this asymptotic expansion to the Newton series for the Dirichlet L-functions. Additional results concern the generalization of bn to b(s) for generalized complex s, so that b(n) = bn at the integers n. Another set of sections explore the relationship to the Stieltjes constants, and attempt to find an asymptotic expansion for these. The development of the sections below are in no particular order; they follow threads of thoughts and ideas in various directions. 2. DERIVATION This section demonstrates the relationship between the Riemann zeta function and the bn, starting from first principles. First, the values of the bn are derived, and then it is shown that the resulting series is equivalent to the Riemann zeta function. The bn may be derived by a straightforward inversion of a binomial transform. Given any sequence {sn} whatsoever, the binomial transform [ref Knuth concrete math] of this sequence is another sequence {tn}, given by n µ ¶ X s t = (−1)k s n k k k=0 The binomial transformation is an involution, in that the original series is regained by the same transform applied a second time: n µ ¶ X s s = (−1)k t n k k k=0 englishNOTES RELATING TO NEWTON SERIES FOR THE RIEMANN ZETA FUNCTION 3 A demonstration of the above is straight-forward, in part because each sum is finite. By assuming the sequence for the Riemann zeta at the integers: n µ ¶ 1 X n ζ(n) = + (−1)k b n − 1 k k k=2 and defining 1 s = ζ(n) − n n − 1 for n ≥ 2, while leaving s0 and s1 temporarily indeterminate, one may invert the series as n µ ¶· ¸ X n 1 b = s − ns + (−1)k ζ(k) − n 0 1 k k − 1 k=2 The harmonic numbers appear as n µ ¶ X n 1 (−1)k = 1 − n + nH k k − 1 n−1 k=2 To complete the process, one need only to make the identification 1 1 b0 = s0 = lim ζ(s) − = ζ(0) + 1 = s→0 s − 1 2 and 1 s1 = lim ζ(s) − = γ s→1 s − 1 Thus, the binomial transform as been demonstrated at the integer values of the zeta func- tion. Next, one must prove that the series ∞ µ ¶ 1 X b s + (−1)n n s − 1 n! k n=0 is identical to the Riemann zeta function for all complex values of s. To do this, one considers the function ∞ µ ¶ 1 X b s f(s) = −ζ(s) + + (−1)n n s − 1 n! k n=0 and note that f(s) vanishes at all non-negative integers. By Carlson’s theorem (given be- low), f(s) will vanish everywhere, provided that it can be shown that f(s) is exponentially bounded in all directions for large |s|, and is exponentially bounded by ec|s|for some real c < π for s large in the imaginary direction. XXX this proof should be supplied. 3. GENERATING FUNCTIONS The ordinary and the exponential generating functions for the coefficients bn are readily obtained; these are presented in this section. englishNOTES RELATING TO NEWTON SERIES FOR THE RIEMANN ZETA FUNCTION 4 Theorem 3.1. The ordinary generating function is given by X∞ n β(z) = bnz n=0 · µ ¶¸ −1 z 1 (3.1) = + ln(1 − z) + ψ 2(1 − z) (1 − z)2 1 − z where ψ is the digamma function. Proof. The above may be obtained by straightforward substitution followed by an ex- change of the order of summation. The harmonic numbers lead to the logarithm via the identity X∞ zk ln(1 − z) = − k k=1 while the binomial series re-sums to the digamma function, which has the Taylor’s series X∞ ψ(1 + y) = −γ + (−1)nζ(n)yn−1 n=2 (equation 6.3.14 from Abramowitz and Stegun[?]). ¤ The above simplifies considerably, to µ ¶ w − 1 w β = + (w − 1)(ψ(w) − ln w) w 2 upon substitution of z = (w − 1)/w. An exponential generating function is given by " # X∞ zn 1 X∞ (−z)k (3.2) b = ez z (1 − γ + Ein(z)) − + ζ(k) n n! 2 k! n=0 k=2 where Ein(z) is the entire exponential integral: X∞ 1 (−z)k X∞ zn (3.3) Ein(z) = − = e−z H k k! n n! k=1 n=1 and Hn are the harmonic numbers. Note. One is left to wonder what the function ∞ µ ¶ 1 X s (3.4) µ(s; x) = + (−)n b xn s − 1 n n n=0 might be like; it is presumably related to the Polylogarithm (Jonquiere’s function) XXXX Do the scratching needed to get the relationship nailed down. Ugh. 4. NUMERICAL ANALYSIS The series expression for the bn is very conducive to numerical analysis. The values of the zeta function may be rapidly computed to extremely high precision using the Tchebysh- eff polynomial technique given by Borwein[?]. Thus, it becomes possible to explore the behavior of the bn up to n ∼ 5000 with only a moderate investment in coding and com- puter time. Note, however, that in order to get up to these ranges, computations must be performed keeping thousands of decimal places of precision, due to the very large values englishNOTES RELATING TO NEWTON SERIES FOR THE RIEMANN ZETA FUNCTION 5 TABLE 1. Some values of bn n bn 0 0.5 1 -0.07721566490153286060651209... 2 -0.00949726295483928474060901... 3 0.001098302680486442197971068... 4 0.001504029222654892992160356... 5 0.000715059124311572276195930... 6 0.000203986913616129918772188... 7 -0.000003382695485446052419661... 8 -0.00005422279825033492206785... This table shows the first few values of bn. These may be easily calculated to high precision if desired. As is immediately apparent, these get small quickly. that the binomial coefficient can take. In the following, calculations were performed using the GNU Multiple Precision Arithmetic Library [?]. The table 1 shows the numerical values of the first few of these constants.
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