some reformulations of the Riemann Hypothesis

The two classic reformulations of the Riemann Hypothesis are "M(x) = o(x1/2 + a) for all a > 0" (WWN notes) and "psi(x) - x = o(x1/2 + a) for all a > 0" (WWN notes).

Here M is a step function satisfying M(0) = 0, and constant except at positive integers, having a jump of $\mu(n)$ at each n. Recall that the Möbius function $\mu(n)$ is defined to be zero if n is divisible by a square, and is otherwise equal to (-1)k where k is the number of distinct prime factors in n.

The function psi(x) is Chebyshev's prime-counting function, which the Prime Number Theorem tells us is asymptotic to x.

Recall that the Landau notation f(x) = o(g(x)) means that limx -> 0 f(x)/g(x) = 0.

The following are all notes from WWN's work-in-progress "Zeta-functions and associated Riemann Hypotheses":

Equivalences to Riemann Hypothesis

The above contains links to abstracts of seven articles which are only accessible to subscribers to MathSciNet. They can be alternatively accessed here:

V.V. Volchov, "On an equality equivalent to the Riemann Hypothesis, Ukr. Math. J. 47 No.3 (1995) 491-493

F. Amoroso, "On the heights of a product of cyclotomic polynomials, Rend. Semin. Mat. Torino 53 No.3 (1995) 183-191

A. Verjovsky, "Discrete measures and the Riemann hypothesis, Kodai Math. J. 17 No.3 (1994) 596-608

J. Alcántara-Bode, "An integral equation formulation of the Riemann hypothesis, Integral Equations Oper. Theory 17 No.2 (1993) 151-168

A. Verjovsky, "Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold", Dyamical Systems. Proceedings of the 3rd international school of dynamical systems, Santiago de Chile, 1990 (R. Bamon,, eds.) Longman Scientific and Tehcnical Pitman Res. Notes Math. Ser 285 (1993) 253-298

W. Barrett,, "On the spectral radius of a (0,1) matrix related to Mertens' function", Linear Algebra Appl. 107 (1988) 151-159

V.M. Popov, "On stability properties which are equivalent to Riemann hypothesis", Libertas Math. 5 (1985) 55-61

RH equivalence to statement involving error term in Prime Number Theorem

RH equivalence to statement involving order of pi(x)-li(x)

M. Riesz's equivalence from the article "Sur l'hypothèse de Riemann", Acta Math. 40 (1916) 185-190

Bombieri's refinement of Weil's positivity criterion

Xian-Jin Li's criterion

G. Robin's equivalence involving the 'sum of divisors' function

Nicolas' equivalence involving Euler's totient function

Massias, Nicolas and Robin's equivalence involving the maximum order of an element in the symmetric group

The Riemann Hypothesis can also be reformulated in terms of a problem involving Farey sequences. This is dealt with in the following:

J. Franel, "Les suites de Farey et les problemes des nombres premiers." Gottinger Nachrichten, 198-201 (1924)

E. Landau, "Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel." Gottinger Nachrichten, 202-206 (1924)

A. Fujii, "A remark on the Riemann hypothesis." Comment. Math. Univ.  St. Pauli 29 (1980), 195-201

A. Fujii, "Some explicit formulae in the theory of numbers. A remark on the Riemann Hypothesis." Proc. Japan Acad., Ser. A 57(1981), 326-330

S. Kanemitsu, and M. Yoshimoto, "Farey series and the Riemann hypothesis." Acta Arith. 75 (1996), no. 4, 351-374

S. Kanemitsu and M. Yoshimoto, "Farey series and the Riemann hypothesis. III." Ramanujan J. 1 (1997), no. 4, 363-378

J. Kopriva, "Contribution to the relation of the Farey series to the  Riemann hypothesis on the zeros of the zeta function (Czech), Casopis Pest. Mat. {\bf 78} (1953), 49-55

J. Kopriva, "Contribution to the relation of the Farey series to the Riemann hypothesis" (Czech), Casopis Pest. Mat. 79 (1954), 77-82

M. Mikolas, "Sur l'hypothese de Riemann." C. R. Acad. Sci. Paris 228 (1949), 633-636

M. Mikolas, "Farey series and their connection with the prime number problem. I." Acta Univ. Szeged. Sect. Sci. Math.13 (1949), 93-117

M. Mikolas, "Farey series and their connection with the prime number problem. II." Acta Univ. Szeged. Sect. Sci. Math.14 (1951), 5-21

M. Mikolas, "On the asymptotic behaviour of Franel's sum and the Riemann hypothesis." Results Math 21(1992) no. 3-4, 368-378

M. Yoshimoto, "Farey series and the Riemann hypothesis. II." Acta Math.
Hungar. 78 (1998), no. 4, 287-304

WWN notes on RH equivalences involving Farey series

G. Mussardo and A. LeClair, "Randomness of Mobius coefficents and brownian motion: Growth of the Mertens function and the Riemann Hypothesis" (preprint 01/2021)

[abstract:] "The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann zeta-function is directly related to the growth of the Mertens function: the RH is indeed true if the Mertens function goes asymptotically as $M(x) \simeq x^{1/2+\epsilon}$. We show that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function. To this aim, we focus the attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdos–Kac theorem for square-free numbers, etc. These results lead us to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk, therefore with an asymptotic behaviour given by $x^{1/2+\epsilon}$. We also argue how the Riemann Hypothesis implies the Generalised Riemann Hypothesis for the Dirichlet $L$-functions. Next we study the local properties of the Mertens function dictated by the Mobius coefficients restricted to the square-free numbers. We perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity. The successful outputs of all these tests (with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive experimental confirmations of the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however ridiculously improbable."

M.L. Lapidus and H. Maier, "Hypothese de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiee", C. R. Acad. Sci Paris Ser. I Math. 313 (1991) 19-24.

(Abstract) "Jointly with C. Pomerance, the first author has recently proved in dimension one the "modified Weyl-Berry conjecture" formulated in his earlier work on the vibrations of fractal drums. Here, we show, in particular, that (still in dimension one) the converse of this conjecture is not true in the "midfractal" case and that it is true everywhere else if and only if the Riemann hypothesis is true. We thus obtain a new characterization of the Riemann hypothesis by means of a inverse spectral problem."

The following contain inter-related reformulations of the RH in terms of scattering theory:

P.D. Lax and R.S. Phillips, Scattering theory (Academic Press, 1967)

P.D. Lax and R.S. Phillips, "Scattering theory for automorphic functions", Bulletin of the American Mathematical Society 2 (2) (1980) 261-295.

L.D. Faddeev and B.S. Pavlov, "Scattering theory and automorphic functions", Proc. Steklov Inst. Math. 27 (1972) 161-193.

The follow purportedly reduces the Riemann Hypothesis to an inverse (quantum) scattering problem, and (despite the humility of the title), claims to contain a proof of the RH.

C. Castro, A. Granik, and J. Mahecha, "On SUSY-QM, fractal strings and steps towards a proof of the Riemann hypothesis" (preprint 07/01)

The following appears to be related to the work of Castro, et. al., insofar as it involves inverse scattering:

N.N. Khuri, "Inverse scattering, the coupling constant spectrum, and the Riemann Hypothesis"

"We use inverse scattering methods, generalized for a specific class of complex potentials, to construct a one parameter family of complex potentials V(s,r) which have the property that the zero energy s-wave Jost function, as a function of s alone, is identical to Riemann's xi function whose zeros are the non-trivial zeros of the zeta function. These potentials have an asymptotic expansion in inverse powers of s(s-1) with real coefficients Vn(r) which are explicitly calculated. We show that the validity of the Riemann hypothesis depends essentially on simple integrability properties of the first order coefficient, V1(r). In the case studied in this paper, this coefficient does not satisfy these conditions, but proof of that fact does indicate several possibilities for proceeding further."

W. Smith, "Cruel and unusual behavior of the Riemann zeta function"

"We exhibit a sequence cn such that the convergence of c1z + c2z2 + c3z3 + ... for |z| < 1 is equivalent to the Riemann Hypothesis. Numerical investigation of the cn revealed some astonishingly deceptive behavior."

J. Lagarias, "An elementary problem equivalent to the Riemann Hypothesis"

J. Sondow, "The Riemann hypothesis, simple zeros and the asymptotic convergence degree of improper Riemann sums", Proceedings of the American Mathematical Society 126 (1998) 1311-1314.

[Abstract:] "We characterize the nonreal zeros of the Riemann zeta function and their multiplicities, using the "asymptotic convergence degree" of "improper Riemann sums" for elementary improper integrals. The Riemann Hypothesis and the conjecture that all the zeros are simple then have elementary formulations."

L. de Branges, "A conjecture which implies the Riemann hypothesis", Journal of Functional Analysis 121 (1994) 117-184.

M.V. Berry has reformulated the Riemann Hypothesis in terms of a search for a dynamical system with a very particular set of properties.

A. Beurling, Proceedings of the National Academy of Sciences 41 (1955) 312.

L. Baez, "On Beurling's real variable reformulation of the Riemann hypothesis", Advances in Mathematics 101 No.1 (1993) 10-30.

E. Saias and M. Balazard, "The Nyman-Beurling equivalent form for the Riemann hypothesis", Expositiones Mathematicae 18 (2) (2000)

J.-F. Burnol, "A lower bound in an approximation problem involving the zeros of the Riemann zeta function"

[abstract:] We slightly improve the lower bound of Baez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the the so called 'Hilbert-Pólya idea'.

J.-F. Burnol, "On an analytic estimate in the theory of the Riemann Zeta function and a Theorem of Baez-Duarte"

[abstract:] "We establish a uniform upper estimate for the values of zeta(s)/zeta(s+A), 0<= A, on the critical line (conditionally on the Riemann Hypothesis). We use this to give a variant, purely complex analytic, to Baez-Duarte's proof of a strengthened Nyman-Beurling criterion for the validity of the Riemann Hypothesis."

L. Baez-Duarte, "A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis" (preprint 02/02)

[abstract:] "Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a | a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $\chi\in\bar{\B}$. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that $\chi\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{\rho_a | a\in\Nat\}$."

L. Baez-Duarte, "Möbius-convolutions and the Riemann hypothesis" (preprint 04/05)

[abstract:] "The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and Hardy-Littlewood, based on the order of growth at infinity along the positive real axis of certain entire functions, are here imbedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion. Some properties of the convolutions involved sharpen what is hitherto known for the Riesz function."

J.-F. Burnol, "An adelic causality problem related to abelian L-functions", Journal of Number Theory 87 no.2 (2001) 253-269.

In this paper, Burnol uses a Lax-Phillips scattering framework to reveal "a natural formulation of the Riemann Hypothesis, simultaneously for all L-functions, as a property of causality."

M. Krishna, "xi-zeta relation", Proceedings of the Indian Academy of Sciences 109 (4) (1999) 379-383

[abstract:] "In this note we prove a relation between the Riemann zeta function and the xi function (Krein spectral shift) associated with the Harmonic Oscillator in one dimension. This gives a new integral representation of the zeta function and also a reformulation of the Riemann hypothesis as a question in L1(R)."

A. Connes, "Formule de trace en geometrie non commutative et hypothese de Riemann", C.R.Sci. Paris, t.323, Serie 1 (Analyse) (1996) 1231-1235.;

(Abstract) "We reduce the Riemann hypothesis for L-functions on a global field k to the validity (not rigorously justified) of a trace formula for the action of the idele class group on the noncommutative space quotient of the adeles of k by the multiplicative group of k."

Berry and Keating refer to this article in their "H = xp and the Riemann zeros", and explain that Connes has devised a Hermitian operator whose eigenvalues are the Riemann zeros on the critical line. This is almost the operator Berry seeks in order to prove the Riemann Hypothesis, but unfortunately the possibility of zeros off the critical line cannot be ruled out in Connes' approach.

His operator is the transfer (Perron-Frobenius) operator of a classical transformation. Such classical operators formally resemble quantum Hamiltonians, but usually have complicated non-discrete spectra and singular eigenfunctions. Connes gets a discrete spectrum by making the operator act on an abstract space where the primes appearing in the Euler product for the Riemann zeta function are built in; the space is constructed from collections of p-adic numbers (adeles) and the associated units (ideles). The proof of the Riemann Hypothesis is thus reduced to the proof of a certain classical trace formula.

relevant articles by Connes and videotaped 1998 lecture series

D. Goldfeld, "Explicit Formulae as Trace Formulae", from Number Theory, Trace Formulas and Discrete Groups (K.E. Aubert, E. Bombieri and D. Goldfeld, eds.) (Academic, 1989) 281-288

"In his epoch-making paper [2], Selberg developed a general trace formula for discrete subgroups of GL(2,R). The analogies with the explicit formulae of Weil [3] (relating very general sums over primes with corresponding sums over the critical zeroes of the zeta-function) are quite striking and have been the subject of much speculation over the years.

It is the object of this note to show that Weil's explicit formula can in fact be interpreted as a trace formula on a suitable space. The simplest space we have been able to construct for this purpose, at present, is the semidirect product of the ideles of norm one with the adeles, factored by the discrete subgroup Q* |X Q, the semideirect product of the multiplicative group of rational numbers with the additive group of rational numbers. We will show that for a suitable kernel function on this space, the conjugacy class side of the Selberg trace formula, is precisely the sum over the primes occuring in Weil's explicit formula.

This implies that the sum of the eigenvalues of the self-adjoint integral operator associated to the aforementioned kernel function is precisely the sum over the critical zeroes of the Riemann zeta-function occurring on the other side of Weil's formula. The relation between the eigenvalues of this integral operator and the zeroes of the zeta-function appears quite mysterious at present. What is lacking is a suitable generalization of the Selberg transform in this situation.

Finally, we should point out that our approach leads to various new equivalences to the Riemann Hypothesis, such as certain positivity hypotheses for the integral operators. Although we have worked over Q, for simplicity of exposition, it is not hard to generalize our results to L-functions of arbitrary number fields."

D. Mayer, "The thermodynamic formalism approach to Selberg's zeta function for PSL(2,Z)", Bulletin of the AMS 25(1991) 55-60. 

"The thermodynamic formalism...leads to a rather explicit representation of the Smale-Ruelle function and hence also of the Selberg function for PSL(2,Z) in terms of Fredholm determinants of transfer operators of the map TG. Finally, combining our results with classical ones for the Selberg function derived from the trace formula suggests also a seemingly new formulation of Riemann's hypothesis on his zeta function in terms of the transfer operators of TG."

According to A. Strombergsson, the following reformulate the RH in terms of problems involving the distribution of closed horocycles:

D. Zagier, "Eisenstein series and the Riemann zeta function", in Automorphic Forms, Representation Theory and Arithmetic, Tata Institute, Bombay (Springer-Verlag, 1981) 275-301.

A. Verjovsky, "Arithmetic geometry and dynamics in the unit tangent bundle of the modular orbifold", Dynamical Systems, Santiago, 1990 (Longman, 1993) 263-298.

Horocycles are most easily understood as horizonal lines in the upper-half-plane model of the hyperbolic plane.

Strombergsson has also pointed out another paper by Verjovsky which reformulates the RH in terms of a 'comparitively elementary equivalence, not involving horocycles':

A. Verjovsky, "Discrete measures and the Riemann hypothesis", Kodai Mathematics Journal 17 no. 3 (1994) 596-608 (Workshop on Geometry and Topology, Hanoi, 1993).

L D Pustyl'nikov, "On a property of the classical zeta-function associated with the Riemann hypothesis", Russ. Math. Surv. 54 (1) (1999), 262-263.

This is an incredibly elegant reformulation of the RH: all even derivatives of the xi function at 1/2 are positive. This is related to some other RH equivalences due to Li, Lagarias and Bombieri.

F. Roesler, "Riemann's hypothesis as an eigenvalue problem. III", Linear Algebra and its Applications 141 (1990) 1-46

[abstract:] "We give conditional induction proofs for the existence of a small zero-free strip inside the critical strip of Riemann's zeta function $\zeta(s)$. The starting point is some formulas for the eigenvalues $\lambda$ of certain matrices AN over the integers, whose determinants are connected with Riemann's hypothesis by the equation $det_{A}N = N!\Sum_{1 \leq n \leq N} \mu(n)/n$, where $\mu$ denotes the Möbius function. The conditions of the proofs refer to properties of the characteristic polynomials XN(x) of the matrices AN near x = 0 and/or the existence of small eigenvalues. A typical example: If for every N > N0 at least one of the polynomials XM(x), $N \leq M \leq N + N^{1 + \epsilon} has a zero $\lambda$ such that -0.09 < $\lambda$ < 1.04, then $\zeta(s) \neq 0$ if $\Re s > 1 - \epsilon$."

D. Merlini, "The Riemann magneton of the primes" (preprint 04/04)

[abstract:] "We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m(r) involving the Zeta function in the complex variable s = r + it and find a particularly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m'(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m (r) gives m = 0 at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1/2, then if we interpret m(r) as a magnetization in the presence of a magnetic field h = r - 1/2 (or as a "free energy" at inverse temperature beta proportional to r), there is a first order phase transition at r = 1/2 (h = 0) with a jump of m'(r) given by 2*Pi times the first Lin coefficient lambda_1 = [1+gamma/2-(1/2)ln(4*Pi)] = 0.0230957. Independently of the RH, by looking at the behavior of the convergent Taylor expansion of m(r) at r = 1-, m(r = 1/2+) as well as the first Lin coefficient may be evaluated using the Euler product formula, in terms of the primes. This gives further evidence for the possible truth of the Riemann Hypothesis."

S. Beltraminelli and D. Merlini, "The criteria of Riesz, Hardy-Littlewood et. al. for the Riemann Hypothesis revisited using similar functions" (preprint 01/06)

[abstract:] "The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of Baez-Duarte concerning a representation of the Riemann Zeta function. Then we introduce a general set of similar functions with the emergence of Poisson-like distributions and we present some numerical experiments which indicate that the RH may barely be true."

M. McGuigan, "Riemann Hypothesis and short distance fermionic Green's functions" (preprint 04/05)

[abstract:] "We show that the Green's function of a two dimensional fermion with a modified dispersion relation and short distance parameter a is given by the Lerch zeta function. The Green's function is defined on a cylinder of radius R and we show that the condition R = a yields the Riemann zeta function as a quantum transition amplitude for the fermion. We formulate the Riemann hypothesis physically as a nonzero condition on the transition amplitude between two special states associated with the point of origin and a point half way around the cylinder each of which are fixed points of a Z2 transformation. By studying partial sums we show that that the transition amplitude formulation is analogous to neutrino mixing in a low dimensional context. We also derive the thermal partition function of the fermionic theory and the thermal divergence at temperature 1/a. In an alternative harmonic oscillator formalism we discuss the relation to the fermionic description of two dimensional string theory and matrix models. Finally we derive various representations of the Green's function using energy momentum integrals, point particle path integrals, and string propagators.


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