The Riemann Hypothesis (RH) is deeply connected to the distribution of prime numbers. Recent work (arXiv:XXXX.XXXXX) on arithmetic progressions offers a novel perspective. This research employs sophisticated Fourier analysis techniques on finite cyclic groups (ℤN) to analyze the distribution of 3-term arithmetic progressions (3-APs).
The core idea is to bridge additive combinatorics (arithmetic progressions, Fourier analysis on ℤN) with multiplicative number theory (prime numbers, the Riemann zeta function). We can achieve this by defining functions on ℤN that encode information about primes and then analyzing their "arithmetic progression profile" using the techniques from the provided paper.
The paper's central equation for counting 3-APs using Fourier coefficients is:
Λ3(f1, f2, f3) = ∑k∈ℤN f̂1(k)f̂2(-2k)f̂3(k) = (1/N)2∑n1,n2,n3∈ℤN, n1+n3=2n2 f1(n1)f2(n2)f3(n3)
This sum quantifies the number of 3-APs (weighted by fi) in ℤN. We propose adapting this framework to analyze functions encoding prime information.
Let f(n) be a function on ℤN representing prime information (e.g., the von Mangoldt function Λ(n) or a characteristic function of primes). The Fourier coefficients are:
f̂(k) = (1/N)∑n=0N-1 f(n)e(-nk/N)
For k coprime to N, these are character sums. Their bounds are crucial in analytic number theory. The distribution of primes in arithmetic progressions directly impacts these sums, which are deeply linked to the zeros of Dirichlet L-functions.
Let fP(n) = Λ(n)/logN for n ∈ {1,...,N} and fP(n)=0 otherwise. Then, as N→∞, the off-diagonal terms of Λ3(fP, fP, fP) are "small" in a specific way. The Riemann Hypothesis implies strong cancellation in sums of Λ(n)e(-nk/N) for k≠0, leading to specific bounds on this sum.
The paper establishes relationships between function bounds and sums of Fourier coefficients. We can leverage these bounds to analyze the "uniformity" of prime distribution. Deviations from uniformity would reveal important information about the distribution of zeros of the Riemann zeta function.