The quantum account and Riemann’s famous hypothesis
Bernhard Riemann (1826-1866) and his teacher Carl Gauss (1777-1855) were pioneers in the development of complex analysis and the topology of curvature in three-dimensional spaces. Let’s explore the famous hypothesis that Riemann proposed in 1859, which is written on the cover of the Cambridge University Press 2012 edition of G.H. Hardy’s A Mathematician’s Apology, and its connection with quantum accounting.
Number meets geometry in the three-dimensional complex plane of the quantum account (part 1)
A complex number is a number that exists in three dimensions, having length, width, and depth and therefore a geometry. Geometry is fundamental to the quantum computer and its transmission of data in the qubit (see Preparing for the Quantum Computing Revolution). The qubit is a sphere, containing a rotating triangle – and so the qubit involves the geometry of triangles and circles and all shapes related to them.
Geometry is not required in accounts that we now recorded in computer bits, because data transmit in straight lines between two opposite states. The logic of today’s computer bits allows for only one of two opposite states (think of them as “on” and “off”) at a time, and produces a single output. The quantum computer qubit, however, will replace the bit, and is very different because it records data in both opposite states simultaneously and therefore produces an unlimited number of potential outputs. In the qubit, opposite states denoted as “1” (“on”) and “0” (“off”) exist simultaneously in the middle of the sphere – and this is what we must account for.
There are two key values to account for in the geometry of the qubit. One of the values is zero, and the other is one-half. Let’s consider each, in relation to today’s accounting and to quantum accounting.
One-Half: In our present accounting, one transaction is recorded in two parts. One half of the transaction is recorded as a credit (belief received, or source of funds) and the other half is recorded as a debit (belief owed, or use of funds). In every transaction, credit always equals debit and so each represents one-half of the transaction and together they total one transaction (since one-half plus one-half equals one).
In the quantum account, every transaction will still be divided into two equal parts – half credit and half debit. The difficulty arises in locating the debit and credit in the spherical qubit. In today’s straight-line computer bits, the line always has a defined middle and each half clearly relates to either debit or credit, depending on the order in the linear sequence.
The problem with spheres is they can never be divided into precise halves. That’s because they are composed of circles, and as everyone knows circles involve π which is the ratio of circumference (curve) to diameter (straight line). π is both irrational and transcendental, and its decimals go on forever – many of us know the first few decimals (π = 3.14159…) but no one, and not even the most powerful computer, has been able to find either an end to or repeating sequence in the ratio of π. Without an end, how can a middle be located? So, when debit or credit can exist either in the circumference or in the diameter of a qubit, how do we measure each in a ratio that goes on – literally, forever?
Zero: In present accounting, zero is the result when a transaction is “in balance” – meaning that there is no difference between debit and credit. The same will be required in quantum accounting, but the problem of spherical geometry in the qubit will make it difficult to locate zero – which is in the middle of the sphere – because how do we find the middle when the ratio of circumference to diameter (in other words, π) never ends?
Consider the other properties of zero. In addition or subtraction, zero makes no difference to the output. Multiply something by zero, however, and the result is always zero. And, of course, we can’t divide by zero which can’t be present in any fraction either as the numerator or as the denominator.
The spherical qubit, which will transmit data to π + π = 360 degrees in the quantum account
Ϛ(s) = 0 => Re(s) = ½
Riemann’s Hypothesis: It’s technical and best left to the mathematicians to define precisely; there is a $1 million prize for a mathematical proof. But in the context of quantum accounting and data in the qubit we can read the left and right sides of Riemann’s Hypothesis as follows:
Ϛ(s) = 0: The Greek symbol (“zeta”) is the term used for the base of the natural logarithm in three dimensions. Accountants will be familiar with zeta’s two-dimensional analogue, e, which is used in the calculation of continuously compounding interest. The base of the natural logarithm is its own derivative, meaning that it is its own source and requires nothing else to sustain it. The necessity for accounting to maintain a balance of 0 in the continuously compounding three-dimensional spherical geometry of the quantum account is the same as the existing requirement for zero difference between debit and credit in our linear accounts.
Re(s) = ½: The Re refers to “real numbers”, which in three dimensions are infinitely dense. There is no end to real numbers; like the decimals of π they go on forever and have no limit. And so it is, in quantum accounting, that there must be no end to our ability to divide every transaction into equal halves credit and debit with no difference – even while it is difficult to locate zero, which is the point of no difference, in the spherical geometry of the qubit.
The fraction ½ has always been and will always be essential to accounting as it moves from two to three dimensions.
In the language of quantum accounting we might think of Riemann’s Hypothesis as follows: “The continuous compounding of a balance of zero in the three dimensions of the account approaches the unlimited capacity of all number of transactions to divide equally into debit and credit.” This will hold for the account in each of three dimensions, just as it now holds for the account in the one dimension of a straight line from its beginning to end.