International Mathematics and Mathematical Sciences Volume 200, Article ID 7447, 9 pages doi:0.55/200/7447 Research Article On Maslana s Representation for the Riemann Zeta Function Luis Báez-Duarte Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado 20632, Caracas 020-A, Venezuela Correspondence should be addressed to Luis Báez-Duarte, lbaezd@cantv.net, lbaezd@gmail.com Received 8 October 2009; Accepted 0 May 200 Academic Editor: Dorothy Wallace Copyright q 200 Luis Báez-Duarte. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. A rigorous proof is given of the hypergeometric-lie representation of the Riemann zeta function ζ s discovered by Maslana as a series of Pochhamer polynomials with coefficients depending on the values of ζ at the positive even integers.. Introduction In, 2 Maslana introduced the following representation of ζ s valid for all s C: s ζ s s ) A P,. 2 0 where the A are given by ) A 2 ) ) ζ 2 2,.2 and the P are the so-called Pochhammer polynomials, defined by P s : s ) r r P 0 s :..3
2 International Mathematics and Mathematical Sciences We apologize for having changed the notation used in, 2, but we have found it more natural to write it as above with future applications in mind. Maslana indicates that he was led to this interesting expression by the desire to interpolate ζ s with a series of the form. treating the A as indeterminates to be found from the system of equations 2n ζ 2n A P n n, 2,.....4 0 This system is a triangular system because P m 0 when the integer m. Its unique solution is found by a nice combinatorial argument given in Appendix A of. In, 2 Maslana gave two formal proofs that the series. with the coefficients as defined by.2 represents s ζ s. However, no estimate of the size of the A is given that would imply the convergence of the series in any region of the complex plane, and which would be needed to ustify several of the formal interchanges of limits and series involved in the proofs. Professor Maslana himself indly indicated this gap to me in a personal communication. I thought it would be worthwhile to devote some effort, which proved quite rewarding, to provide the missing steps in the proof. An effective estimate of the rate at which A 0 is the crucial missing lin in the proof, and to this effect we shall show below that A p p.5 for every positive real p. In closing this introduction, we would lie to point out an interesting paradox that could deserve some attention. Once the Maslana representation is shown to be valid in the whole complex plane it is obvious that. will be valid for s, 0,, 2,...;noting however that in this case the series does not truncate, one obtains interesting identities as pointed out in. If, on the other hand, one proposed at the outset the analogous problem of finding a representation of the form s ζ s c P 2 s, 0.6 then, since it is clear that the series truncates for s, 0,, 2,...one gets again a triangular system leading to a surprising solution, namely, s ζ s 2 s B P 2 s,.7 where the B are the Bernoulli numbers. This happens to be, in different garb, the divergent Euler-Maclaurin series of ζ s. It may be obected that Dirichlet series albeit s ζ s is not a Dirichlet series are determined by its values at any sequence with real parts tending to, while that is not the case toward the left. It may seldom be the case that this question is raised at all since all Dirichlet series of interest have a finite abscissa of convergence so there is no way to move toward. On the other hand, Professor Malansa claims he has
International Mathematics and Mathematical Sciences 3 a representation that interpolates simultaneously at points 2n and 3 2n. We do not now what that series may be, perhaps he means something lie c P s/2 P s /2. These considerations can be generalized to explore general p-series c P s which share certain characteristics with Dirichlet series. As an example we could find a p-series for ζ s and arrived at the following criterion for the Riemann hypothesis that involves the Bernoulli numbers: define b : ) ζ ) 0,, 2,.....8 2 2 Then the Riemann hypothesis is true if and only if b ɛ 3/4 ɛ ɛ >0..9 This criterion is a discrete version the author published in 3 of the well-nown Riesz criterion for the Riemann hypothesis 4. It is well suited for numerical calculations. Plotting the results of some preliminary computations, one obtains a very smooth curve b 4/3 that quite impresses one as tending to zero lie log 2. 2. Statement and Proof of the Main Theorem We now formally state and prove the representation theorem. Theorem 2.. For A defined as in.2, we have s ζ s s ) A P, 2. 2 for all s C. The convergence of the series is uniform and absolute in every compact set of the complex plane. Throughout, it shall be very important to bear in mind the simple estimate for the size of the Pochhammer polynomials contained in the following lemma already proved in 3, Lemma 2.3. Lemma 2.2. For every compact set H C, there is a positive constant C H, not depending on,such that P s C H Rs s H,, 2,.... 2.2 Do note also the equally trivial facts contained in the following lemma Lemma 2.3. ) P s s Γ s!γ s. 2.3
4 International Mathematics and Mathematical Sciences Proof of Theorem 2.. The ey estimate.5 for the coefficients A shall be proved in the next section. We accept it for the moment and show now how the informal steps in Maslana s second proof can be made rigorous. We shall first show that with no assumption on the size of the A one can prove equality in. for Rs >4. We start with Maslana s clever idea of expressing s as the derivative of α s at α. We indicate by D α the operation of differentiating with respect to α and proceed as follows: α s s ζ s D α α s n n s D α α αn s n D α α αn 2 αn 2 n ) s/2 D α α αn 2 αn 2 n n )) s/2 s D α α αn 2 2 ) αn 2 D α n 0 0 s P 2 ) αn 2 αn 2 ), ) 2.4 where we have applied the binomial theorem assuming as we may that the derivative is calculated from the right so α>. Consider now the formally differentiated double series on the right-hand side above, namely, n 0 ) ) s ) 2 P. 2.5 2 αn 2 αn 2 αn 2 Using Lemma 2.2, we see that this double series is compact-uniformly termwise maorized by a convergent double series of positive terms in Rs 4 2ɛ since s ) P 2 αn 2 ) 2 αn 2 αn 2 ) ɛ n 2. 2.6 Thus we can both invert the sums at the end of 2.4 and do the differentiation termwise at α obtaining s ζ s s ) P 2 0 n n 2 ) 2 ). n 2 n 2 2.7
International Mathematics and Mathematical Sciences 5 We easily calculate the last series on the right as ) ζ ) )) 2 2 2 ζ 2 4 2.8 which, after separating in two sums and shifting indices in the second, becomes ) 2 ) ) ζ 2 2, 2.9 which is none other than A. One could even argue this would be a direct way to find the coefficients. We have thus shown the desired relation. at least for Rs >4. However, the strong estimate.5, to be proved below, providing that A p for every p>0, together with Lemma 2.2, immediately shows that the series A P s/2 converges uniformly on any compact subset of the plane, thus defining an entire function that must be equal to s ζ s by analytic continuation. 3. The Size of the Coefficients We shall establish here this Proposition which was announced as the essential estimate.5 in the introduction. Proposition 3.. For any p>0 there is a constant C p > 0 such that A C p p,, 2,... 3. To lighten up the proof, we first prove some lemmas. Define the crucial sequence of rational functions φ x by φ x : x 2 ) x Z. 3.2 We now record a very simple while extremely important property of φ x. Lemma 3.2. For 0 a, denote φ a da x : dx a φ x. 3.3 Then we have φ a φ a 0, 3.4 where the second equality is unrestricted on a. Alltheφ a x for a are integrable.
6 International Mathematics and Mathematical Sciences The following lemma is the essential and quite elementary tool to get a grip on the size of A. Lemma 3.3. Define a double sequence of polynomials p a,, with integral coefficients as follows: let p 0,0, p a, 0 when <0 and when >a, and specify the recurrence equation p a, 2 a ) p a, 2 2 a ) p a,. 3.5 Each p a, is a polynomial of degree. For any integers a 0, φ a x ) a a x 2 p a, x. a 2 3.6 The simple proof is by double induction. The recurrence relation 3.5 will not be used other than the fact that it is part of the proof. It is nice to record it explicitly for further use if need be. Having straightened some notation, as all the above was essentially, we are ready for a first preliminary estimate. Lemma 3.4. For any fixed integer a and ɛ>0, there is a constant C C a, ɛ such that φ a x dx C a/2 ɛ. 3.7 Proof. Tae a. We split the interval of integration at the point x /2 ɛ/3. We express the ath derivative according to Lemma 3.3 and equation 3.6. For the finite range of the integral, we have /2 ɛ/3 φ a dx /2 ɛ/3 x ) a a p a, x 2 x a 2 dx a /2 ɛ/3 p a, ) a dx x 2 O a a 2ɛ/3 ) a /2 ɛ/3 3.8 O a e ɛ/3).
International Mathematics and Mathematical Sciences 7 Now for the infinite range of integration we have φ a /2 ɛ/3 dx x ) a a p a, /2 ɛ/3 x 2 x a 2 dx a pa, a a 2 a /2 ɛ/3 a /2 ɛ/3 dx /2 ɛ/3 xa 2 pa, /2 ɛ/3 a 2 a a 2 a a 2 O a pa, 2ɛ/3 2ɛ/3 ) O a /2 ɛ/3 a 2ɛ/3 a) O a a/2 ɛ/2 ). 3.9 Since this last estimate 3.9 surely dominates the smaller one in 3.8, we can allow the substitution of ɛ/2byɛ to arrive at the desired 3.7. Remar 3.5. Actually there is no need for ɛ to be arbitraily small for the use this estimate 3.7 is destined for. If one taes, say, ɛ we see that the order obtained is O a a/4, which is plenty, since we are planning to differentiate a large but fixed number of times, while. The idea perhaps was to see how much we can ee out of this method, which is as close as a/2 as desired. Perhaps a more painstaing analysis, taing into account the actual nature of the polynomials p a,, could yield an essentially faster order of convergence to zero, which is desirable for the effectiveness of the Maslana representation. At this point it is high time to connect the coefficients A with the function φ.sowe have first a high school triviality, namely, the following lemma. Lemma 3.6. For any x / 0 ) 2 ) x 2 2 φ x. 3.0 Proof. ) 2 ) d ) x2 2 dx x x 2 d x ) ). dx x 2 3.
8 International Mathematics and Mathematical Sciences It should now be clear why the wor on the derivatives of φ x is carried out if the reader notices that the sum below should be estimated by Euler-Maclaurin with a large number of terms. Lemma 3.7. A φ n. n 3.2 Proof. The following interchange of sums is totally elementary to ustify: ) A 2 ) ) ζ 2 2 ) 2 ) n φ n, n n n 2 2 ) 2 ) n 2 2 3.3 where, of course, we applied Lemma 3.6 in the last equality. We are finally ready to prove the estimate for the A. Proof of Proposition 3.. Tae an integer a > 4p and assume that > a; naturally one will mae. The crucial elementary properties of φ and its derivatives expressed in Lemma 3.3 imply the remarable fact that the application of the Euler-Maclaurin summation formula to a depth of a steps to the sum in 3.2, thatis,to A φ n n 3.4 results in the sum being equal to the remainder term! Therefore, A a a! B a x φ a x dx, 3.5
International Mathematics and Mathematical Sciences 9 where B a x is the ath periodified Bernoulli polynomial. Now apply Lemma 3.4 and equation 3.7 with ɛ /2 toget A B a a! φ a x dx 3.6 O a a/4) O a p ). Remar 3.8. We believe there is a room for improvement in estimating the smallness of the A. This should have a bearing on how useful the Maslana formula may turn out to be in the end. References K. Maslana, A hypergeometric-lie Representation of Zeta-function of Riemann, Cracow Observatory preprint no. 997/60, July 997. 2 K. Maslana, Hypergeometric-lie representation of the zeta-function of Riemann, http://arxiv.org/abs/math-ph/005007. 3 L. Báez-Duarte, A sequential Riesz-lie criterion for the Riemann hypothesis, International Mathematics and Mathematical Sciences, no. 2, pp. 3527 3537, 2005. 4 M. Riesz, Sur l hypothèsederiemann, Acta Mathematica, vol. 40, no., pp. 85 90, 96.
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