Criteria equivalent to the Riemann Hypothesis

for real s > 1. In particular, he found expression giving values of the zeta function for even arguments:

where B n are the Bernoulli numbers. Riemann showed [23] that the integral (s = 1):

where the integration is performed on the contour , see e.g. [27]. The Riemann Hypothesis (RH) states that all non-trivial zeros of ζ(s) lie on the critical line s = 1 2 + it. Presently the requirement that they are simple is often added. It is one of the best known open problems in mathematics, see e.g. [8], [9]. In the last years, after the Clay Mathematics Institute granted 1 million US$ award for solving the dilemma of the RH, there have appeared on the arxiv many preprints claiming to have proved (e.g., [25], [14]) or disproved (e.g. [1], [21]) the RH, but so far all of them have been withdrawn as errors were found in them.

Already Riemann calculated numerically a few first nontrivial zeros of ζ(s) [9]. Below there is a short list of numerical determinations of nontrivial zeros of ζ(s):

J.P. Gram(1903): 15 zeros are on the critical line [11] . . . A. Turing (1953): 1104 zeros are on the critical line [28] . . . A few years ago S. Wedeniwski (2005) was leading the internet project Zetagrid [31] which during four years determined that 250 × 10 12 zeros are on the critical line: s = 1 2 + it, |t| < 29, 538, 618, 432.236. The present record belongs to K. Gourdon(2004) [10]: the first 10 13 zeros are on the critical line.

There are probably well over one hundred statements equivalent to the RH, see eg. [27], [12], [30]. Riemann’s original aim was to prove the guess made by 15-yearsold Gauss, namely that the number π(x) of primes < x is well aproximated by the logarithmic integral:

In this spirit in 1901, Koch proved [29] that the Riemann Hypothesis is equivalent to the following error term for the expression for the prime counting function: π(x) is given by: π

Another similar criterion is

The following criterion is of interest to mathematical physicists. In 1955 Arne Buerling [3] proved that the Riemann Hypothesis is equivalent to the assertion that N (0,1) is dense in L 2 (0, 1). Here N (0,1) is the space of functions

where ρ(u) = u -u is a fractional part of u. The function ζ(s) does not have zeros in the half-plane σ > 1 q , 1 < q < ∞ iff the set N (0,1) is dense in L q (0, 1). In fact Beurling proved that the following three statements regarding a number q ∈ (1, ∞) are equivalent:

(1) ζ(s) has no zeros in σ > 1/q (2) N (0,1) is dense in L q (0, 1)

3. The characteristic function χ (0,1) is in the closure of N (0,1) in L q (0, 1)

The following ideas show that the validity of the RH is very delicate and subtle. Let us introduce the function

We can see from the above formula that the RH ⇔ all zeros of ξ(iz) are real. The point is that ξ(z) can be expressed as the following Fourier transform:

where

And now we follow the rule of Polya [22] : if one can not solve a particular problem, maybe it is possible to solve more general problem. So, we introduce the family of functions H(z, λ) parameterized by λ as the following Fourier transform:

Thus we have H(z, 0) = 1 8 ξ( 1 2 z) N. G. De Bruijn [7] proved that (1950): 1. H(z, λ) has only real zeros for λ ≥ 1 2 2. If H(z, λ) has only real zeros for some λ , then H(z, λ) has only real zeros for each λ > λ . And here comes bad news: in 1976 Ch. Newman [19] has proved that there exists parameter λ 1 such that H(z, λ 1 ) has at least one non-real zero. Thus, there exists such constant Λ in the interval -∞ < Λ < 1 2 that H(z, λ) has real zeros ⇔ λ > Λ. The Riemann Hypothesis is equivalent to Λ ≤ 0. This constant Λ is now called the de Bruijn-Newman constant. Newman believes that Λ ≥ 0. The computer determination has provided the numerical values of de Bruijn-Newman constant, here is a sample of results:

Because the gap in which Λ catching the RH is so squeezed, Odlyzko noted in [20], that “…the Riemann Hypohesis, if true, is just barely true”. Li Criterium (1997) [15]: Riemann Hypothesis is true iff the sequence:

Explicit expression: [18], [17] gave explicit expression for λ i and performed extensive computer calculations of these constants confirming (13).

A sensation was stirred in 2000, when the elementary criterion for the RH was invented by Lagarias [13]: the Riemann Hypothesis is equivalent to the inequalities:

for each n = 1, 2, . . ., where H n is the n-th harmonic number H n = n j=1 1 j . Here σ(n) is the sum of all divisors of n. The illustration of ( 14) is shown on Fig. 1. In the paper [4] the maxima of σ(n) were studied. Fig. 1 The plot of σ(n) for 1 < n < 10 6 . In red are plotted values of σ(n) which approach the threshold values closer than 5%. Data for this plot was obtained with the free package PARI/GP [26] 3. Criteria of Riesz and Báez-Duarte

In 1916 Riesz [24] introduced the function:

and next he proved that:

It involves uncountably many values of x and L. Báez-Duarte [2] found a discrete version of (16). I