The Riemann hypothesis: Generalizations of the Riemann hypothesis

For any given mathematical statement, whether it is an established theorem or an unproven conjecture, there is almost always some way to make it “stronger”. One way to make a statement stronger is to make its conclusions more precise. For instance, the error bounds in an approximation might be made tighter. Another way to make a statement stronger is to make its assumptions less precise, i. e, weaker. For instance, a theorem about analytic functions could be strengthened if it could also be shown (perhaps with modifications) to apply to functions that have singularities.

Similar to strengthening a statement by weakening the assumptions is to make the results more general, i. e. the assumptions on objects that the statement applies to are made less restrictive. For instance, statements about integers can frequently be strengthened by applying them to more general objects, such as groups or rings, which do not have the full range of properties that the integers do. As we shall see, there are conjectures very similar to the Riemann hypotheses that apply to a wide variety of functions other than the zeta function, but analogous to it in some way.

We’ve already noted that it isn’t possible to strengthen one sort of result which is equivalent to the Riemann hypotheses – namely with respect to the size of the error estimate in the prime number theorem. Results of that sort pertain to the number theoretic functions π(x) and ψ(x). There could easily be other number theoretic functions in terms of which one can state results which are either stronger than the Riemann hypothesis (in other words, generalizations) because they imply it, or weaker because they are implied by it. (In addition to the possibility of being equivalent.)

What are some other interesting number theoretic functions? Well, there’s the Möbius function μ(n) which has been mentioned already and is defined for integers by μ(n) = 0 if n is divisible by the square of a prime, μ(n) = 1 if n is a product of an even number of distinct primes, and μ(n) = -1 if n is a product of an odd number of distinct primes. Now, whenever a number theorist deals with functions defined on (positive) integers, one of the first impulses is to think about the Dirichlet series, which is one type of “generating function” for the given sequence of numbers. That is, a function defined formally by ∑n an n-s that corresponds to the sequence {an}.

So we might well wonder about the function defined by ∑n μ(n) n-s. Would you be shocked to learn that this function is precisely the reciprocal of ζ(s)?

1/ζ(s) = ∑1≤n<∞ μ(n)/ns

Just like ζ(s) this series converges and gives a holomorphic function for Re(s) > 1. If ζ(s) had any zeros for 1/2 < Re(s) ≤ 1, 1/ζ(s) wouldn’t be a holomorphic function in that region. The Riemann hypothesis says there are no such zeros. In fact, assuming the Riemann hypothesis, it can be shown that not only is 1/ζ(s) holomorphic for Re(s) > 1/2, but the Dirichlet series converges in that region.

We recall the very similar series

ζ′(s)/ζ(s) =-∑1≤n<∞ Λ(n)/ns

which played such a crucial role in the prime number theorem. Just as we had ψ(x) = ∑0<n≤x Λ(n), we cannot help but wonder whether the function defined by

M(x) = ∑0<n≤x μ(n)

might be very interesting.

It is, and just as with ψ(x), one of the most interesting things about it is its asymptotic size. It can be shown that the estimate

M(x) = O(x1/2 + ε) for any ε > 0

is equivalent to the Riemann hypothesis.

Even more could conceivably be true. On the basis of numerical evidence at the time, Franz Mertens (whose theorem concerning Euler’s constant γ and the Γ function has already been mentioned) conjectured that in fact |M(x)| < √x for all x. This is known as the Mertens conjecture. It would imply the Riemann hypothesis, if true. However, it’s actually a stronger statement, and therefore a generalization of the Riemann hypothesis.

But in this case this is a plausible conjecture that has been disproven. In 1984 A. M. Odlyzko and H. J. J. te Riele proved that for some values of x, M(x)/√x > 1.06, and for others M(x)/√x < -1.09. The proof was indirect and didn't actually exhibit any x satisfying those inequalities. It is still unknown whether M(x) = O(√x), though that seems doubtful. The negative conclusion to Mertens' conjecture shows one can't always depend on extensive numerical evidence in formulating number theoretic conjectures. This rather reduces one's enthusiasm for believing the Riemann hypothesis based solely on numerical evidence. (A similar failure occurred with the conjecture that π(x) < Li(x), which was known to be true for x up to 3 million. But it was later proven that the quantity Li(x) – π(x) changes sign infinitely often.)

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