Effective Results in Number Theory - Littlewood's Result

Littlewood's Result

An early example of an ineffective result was J. E. Littlewood's theorem of 1914, that in the prime number theorem the differences of both ψ(x) and π(x) with their asymptotic estimates change sign infinitely often. Until the result on the Skewes number of 1933, these results were believed by some experts to be intrinsically ineffective.

In more detail, writing for a numerical sequence f(n), an effective result about its changing sign infinitely often would be a theorem including, for every value of N, a value M > N such that f(N) and f(M) have different signs, and such that M could be computed with specified resources. In practical terms, M would be computed by taking values of n from N onwards, and the question is 'how far must you go?' A special case is to find the first sign change. The interest of the question was that the numerical evidence known showed no change of sign: Littlewood's result guaranteed that this evidence was just a small number effect, but 'small' here included values of n up to a billion.

The requirement of computability reflects on and contrasts with the approach used in analytic number theory to prove the results. It for example brings into question any use of Landau notation and its implied constants: are assertions pure existence theorems for such constants, or can one recover a version in which 1000 (say) takes the place of the implied constant? In other words if it were known that there was M > N with a change of sign and such that

M = O(G(N))

for some explicit function G, say built up from powers, logarithms and exponentials, that means only

M < A.G(N)

for some absolute constant A. The value of A, the so-called implied constant, may also need to be made explicit, for computational purposes. One reason Landau notation was a popular introduction is that it hides exactly what A is. In some indirect forms of proof it may not be at all obvious that the implied constant can be made explicit.