Some Riemann Hypotheses from Random Walks over Primes
The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of -functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is , then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results we obtain an equation which relates every individual non-trivial zero of the -function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet -functions due to the existence of the pole at , in which the Riemann -function is a particular case.
Montgomery conjectured that the pair correlation function between the ordinates of the Riemann zeros on the critical line satisfy the GUE statistics of random matrix theory Montgomery. On the other hand, Riemann Riemann obtained an exact formula for the prime number counting function in terms of the non-trivial zeros of . This suggests that if the Riemann Hypothesis is true, then this should imply some kind of randomness of the primes. It has been remarked by many authors that the primes appear random, and this is sometimes referred to as pseudo-randomness of the primes Tao.
In this article we address the following question, which is effectively the reverse of the previous paragraph. What kind of specific pseudo-randomness of the primes would imply the Riemann Hypothesis? This requires a concrete characterization of the pseudo-randomness. We provide such a characterization by arguing that certain deterministic trigonometric sums over primes, involving multiplicative functions, behave like random walks, namely grow as . However, we are not able to fully prove this growth, and thus we will take it as a conjecture. This conjecture may appear to be reminiscent of Merten’s false conjecture that , where is the Möbius function. However, it is different in an important manner: our series involves a sum over primes rather than integers which in some sense renders it more random.
The main result of this paper can be stated as follows. Consider -functions based on non-principal Dirichlet characters and on cusp forms. We prove that, assuming the claim from the previous paragraph concerning the random walk behavior, the Euler product converges to the right of the critical line.
This article is partly based on the ideas in EulerProduct and is intended to clarify it with more precise statements. There is an important difference between the cases mentioned above and principal Dirichlet -functions, where is a particular case, and this is emphasized more here. We will not consider this latter case in detail, but we briefly mention these subtleties in the last section of the paper.
Ii On the growth of series of Multiplicative Functions over primes
In this section we consider the asymptotic growth of certain trigonometric sums over primes involving multiplicative arithmetic functions. We propose that these sums have the same growth as one-dimensional random walks.
Let be a multiplicative function, i.e. and if and are coprime integers, and let denote an arbitrary prime number. We can always write . Now consider the trigonometric sum
where denotes the th prime; , , and so forth. We wish to estimate the size of this sum, specifically how its growth depends on .
ii.1 Non-Principal Dirichlet Characters
ii.1.1 The Main Conjecture
Now let be a Dirichlet character modulo , where is a positive integer. The function is completely multiplicative, i.e. , for all , and obeys the periodicity . Its values are either , or if and only if is coprime to . For a given there are different characters which can be labeled as . The arithmetic function is the Euler totient. We will omit the index of the character except for which denotes the principal character, defined as if is coprime to , and otherwise. The Riemann -function corresponds to the trivial principal character with .
For a non-principal character the non-zero elements correspond to -th roots of unity given by for some . The distinct phases of these roots of unity form a discrete and finite set denoted by
Here is the order of the character. For prime, .
For our purposes, there is an important distinction between principal verses non-principal characters. The principal characters satisfy
while non-principal characters satisfy
The above relation (4) shows that the angles in are equally spaced over the unit circle for non-principal characters. On the other hand, this is not the case for principal characters due to (3); in fact the angles are all zero.
For the sake of clarity, let us now simply state the main hypothesis that the remainder of this work relies upon. We cannot prove this conjecture, however we will subsequently provide supporting, although heuristic, arguments.
Let be the th prime and the value of a non-principal Dirichlet character modulo . Consider the series
Then as , up to logs. By the latter we mean, for instance, for any positive power , or , etc. suffices.
The main supporting argument is an analogy with one-dimensional random walks, which are known to grow as . Although the series is completely deterministic, its random aspect stems from the pseudo-randomness of the primes, which is largely a consequence of their multiplicative independence. The event of an integer being divisible by a prime and also divisible by a different prime are mutually independent. A simple argument is Kac’s heuristic Kac: let denote the probability that an integer is divisible by . The probability that is even, i.e. divisible by , is . Similarly, . We therefore have , and the events are independent. Because of the multiplicative property of this independence of the primes extends to quantities involving , in that is independent of for primes . Moreover, if are equidistributed over a finite set of possible angles, then the deterministic sum (1) is expected to behave like a random walk since each term mimics an independent and identically distributed (iid) random variable. Analogously, if we build a random model capturing the main features of (1) it should provide an accurate description of some of its important global properties.
Let us provide a more detailed argument. First a theorem of Dirichlet addresses the identically distributed aspect.
Theorem 1 (Dirichlet).
Let be a non-principal Dirichlet character modulo and the number of primes less than . These distinct roots of unity form a finite and discrete set, with . Then for a prime we have
for all , where denotes the frequency of the event occurring.
Let denote the residue classes modulo for and coprime, namely the set of integers . There are independent classes and they form a group. Of these classes let the set of integers denote the particular residue class where . Then
Dirichlet’s theorem states that there are an infinite number of primes in arithmetic progressions, and independent of . In particular,
in the limit . (See for instance (Davenport, Chap. 22).) ∎
The frequencies can be interpreted as probabilities, however we will continue to refer to them as frequencies. Next consider the joint frequency, defined by
for all . The events and ( are independent due to the multiplicative independence of the primes. Thus one expects
In other words, for a randomly chosen prime, each angle is equally likely to be the value of , i.e., is uniformly distributed over . Moreover, and are independent. Thus the series (5) should behave like a random walk, and this is the primary motivation for Conjecture 1.
In ALCLT one of us studied a probabilistic model for and proved a central limit theorem for it. Namely, in the definition of , the primes were replaced with where the were chosen according to Cramér’s random model for the primes Cramer. is now a random variable with a probability distribution, which we showed to be a normal distribution as . The latter implies for any with probability equal to . Also, the law of iterated logarithm implies , which as stated in Conjecture 1, will be sufficient for our purposes.
ii.1.2 Numerical Evidence
Let us also provide numerical evidence for the above statements. In Figure LABEL:fig:proba we have an example with . The specific character is
with , so that . This table was computed with in (6). One can see the equally spaced angles over the unit circle, and the numerical results verify that is uniformly distributed over , as stated in Theorem 1.
Let us also check (9). All the joint frequencies are shown in the following matrix: