RIEMANN’S ZETA FUNCTION: MORE COMPUTATIONS AND CONJECTURES YURI MATIYASEVICH Abstract. In 2007 the author proposed a new approach to the Riemann Hypothesis based on our precise knowledge of positions of the trivial zeros of Riemann’s zeta function. Numerical computations revealed interesting pictures and allowed the author to state new conjectures about the zeta function. The paper presents further computations and a new conjecture.
Introduction In [5, 6] (or, equivalently, see [7]; for future development be visiting [8]) the author proposed a new approach to the Riemann Hypothesis (RH) based on our precise knowledge of the positions of the trivial zeros of Riemann’s zeta function. This approach consists in the study of eigenvalues of special matrices. The Riemann Hypothesis can be reformulated as a statement about the limiting behavior of the geometrical means of the eigenvalues of these matrices. Numerical calculations show interesting visual patterns in the distribution of eigenvalues of such matrices in small dimensions which allowed the author to state a number of conjectures about the zeta function. However, small dimensions might be deceptive and the pictures could drastically change with the growth of dimension. So calculations were continued1 and this paper presents some new numerical evidence in favour of previously stated conjectures. As a natural first step to RH the author suggested to examine eigenvalues of similar matrices constructed for the function which has the same “trivial” zeroes as ζ(s) but has no other zeros. In this paper yet further simplification of the problem is studied: we consider eigenvalues of matrices constructed for a one-parameter family of functions having only one zero. To make this paper self-contained, a brief digestion of [5] is given here. 2000 Mathematics Subject Classification. Primary 11M06; Secondary 11M26. Key words and phrases. zeta function, Riemann Hypothesis, trivial zero. This work was partially supported by the Council for Grants of the President of the Russian Federation under grant NSh-4392.2008.1. 1 The author is grateful to S. Ananko and S. Permyakov for performing part of the computational work.
1. Notation Let z1 , z2 , . . . be an infinite or a finite sequence of distinct negative numbers enumerated in such a way that 0 > z1 > z2 > . . . We are to study the following property which an entire function f (z) can have or have not: Property R(f ). In the left half-plane 1 the limiting target Targl (ζ ∗ ) consists of some number targl (ζ ∗ ) of limiting orbits Orbl,k (ζ ∗ ) and 1I0 all limiting orbits are circles; ∗ 1I00 on each limiting orbit the limiting measure λζl (w) is constant;
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Figure 3. Spectrum Specλ50,999 1I000 the limiting orbits Orbl,k (ζ ∗ ) can be numbered in such a way that for k < targl (ζ ∗ ) the limiting orbit Orbl,k (ζ ∗ ) lies inside the limiting orbit Orbl,k+1 (ζ ∗ ) and touches it at one real point Rendl,k (ζ ∗ ) called the rendezvous-point; the innermost limiting orbit has rendezvous point Rendl,0 (ζ ∗ ) with the limiting arrow Arrl (ζ ∗ ), and the outmost limiting orbit has rendezvous point Rendl,targl (ζ ∗ ) (ζ ∗ ) with the limiting bow Bowl (ζ ∗ ); moreover, Rendl,k+1 (ζ ∗ ) < Rendl,k (ζ ∗ ) for even k and Rendl,k+1 (ζ ∗ ) > Rendl,k (ζ ∗ ) for odd k. It is more difficult to guess the form of the limiting bow. Figures 3–6 show spectrum Specλ50,999 (ζ ∗ ) in several scales. It can be seen that while the innermost orbits are almost circles, the outer orbits are rather distorted; nevertheless, they, as well as the bow, might approach circles with growing dimension of matrices.
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Figure 4. Part of spectrum Specλ50,999
3. The case of trivial zeros The author also calculated spectra of the “trivial” function z z ζT (z) = π − 2 Γ(1 + ) 2
(14)
with the same real zeros as ζ ∗ (z) but having no other zeros. Visually spectra Specλl,m (ζ ∗ ) and Specλl,m (ζT ) for small l and m look very much alike (see [5, 6, 7, 8]). This suggests the following possible approach to RH: at first study the eigenvalues of matrices Ll,m (ζT ) and then treat the eigenvalues of matrices Ll,m (ζ ∗ ) as perturbations of the former eigenvalues. Here we go further in this direction: we are to study the case of a function with a single negative zero. Let a be a positive number. We take z1 = −a−1 and consider function φa (z) = 1 + az in the role of function f in the previous development.
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Figure 5. Part of spectrum Specλ50,999 Respectively, az −1 φ˜a (w) = 1 + , w1 = , W1 = a + 1, z+1 1+a expansion (9) takes the form φ˜a (z) = 1 − aw + aw2 − aw3 + . . . and the matrix L1,m (φa ) looks like a 1 0 −a a 1 a −a a . .. .. .. . . ±a ∓a ±a ∓a ±a ∓a
... ... ... .. .
0 0 0 .. .
0 0 0 .. . 1
(15)
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... a . . . −a a It is easy to check that det L1,m (φA a) = a(a + 1)m−1 , in other words, a . condition (10) holds with R1 (φa ) = a+1
Moreover, an explicit expression for the characteristic polynomial of the matrix L1,m (φa ) can be given: m X m−i X i+j i j m m j m−j −1 aλ. (17) (−1) λ + (−1) i − 1 i i=1 j=0 This leads to the following formal series expression for the eigenvalues: k−1
∞ X m X 2(−1)j+k (k + 1)(jm − k)Γ(k + k=1 j=0
mΓ(1 + j + 1)Γ(1 +
2k m
2k m
− j − jm − 1) k b − j)Γ(k − j − jm + 2) m
(18)
where bm is any of the m solutions of the equation (−b)m = −a. The summands in the inner sum in (18) cancel heavily, and it is not clear whether the series actually converges. Numerical calculations of Specλ1,m (φa ) indicate that it consists of a “bow” only and allows one to state the following conjecture: Conjecture 1F1 (φa ). Assign the weight m1 to each of the points a λ1,m,1 (φa ), λ1,m,2 (φa ), . . . , λ1,m,m (φa ) and denote by λφ1,m corresponding discrete measure. Then there exists a limiting continuous measure λφ1 a (w) concentrated on the arc |w| = W1 = a + 1, a − 1. References [1] Baker G. A., Jr. Essntials of Pad´e Approximations. Academic Press, New York, SanFrancisco, London, 1975. [2] de Montessue de Ballore. Sur le fraction continues alg´ebriques. Bull. Soc. Math. France, 30, 28–36, 1902. [3] de Montessue de Ballore. Sur le fraction continues alg´ebriques. Rend. Circ. Math. Palermo, 19, 1–73, 1905. [4] Jacobi C. G. J. Uber die Darstellung einer Reihe Gegebner Werthe durch eine Gebrochne Rationale Function. J. Reine Angew. Math., 30, 127–156, 1846. [5] Matiyasevich, Yu. Hidden Life of Riemann’s Zeta Function 1. Arrow, Bow, and Targets. http://arxiv.org/abs/0707.1983, 2007. [6] Matiyasevich, Yu. Hidden Life of Riemann’s Zeta Function 2. Electrons and Trains. http://arxiv.org/abs/0709.0028, 2007. [7] Matiyasevich, Yu. Riemann’s zeta function: Some computations and conjectures. In A.-M. Ernvall-Hytonen, M. Jutila, J. Karhum¨aki, and A. Lepsito, editors, Proceedings of Conference on Algorithmic Number Theory, TUCS General Publication Series, volume 46, 87–112, 2007. [8] Matiyasevich, Yu. Hidden Life of Riemann’s Zeta Function. http://logic. pdmi.ras.ru/~yumat/personaljournal/zetahiddenlife. Steklov Isnstitute of Mathematics at St.Petersburg, Fontanka, 27, St.Petersburg, 191023, Russia E-mail address:
[email protected] URL: http://logic.pdmi.ras.ru/~yumat
