Exponential Sums and the Distribution of Prime Numbers

We study growth estimates for the Riemann zeta function on the critical strip and their implications to the distribution of prime numbers. In particular, we use the growth estimates to prove the Hoheisel-Ingham Theorem, which gives an upper bound for the difference between consecutive prime numbers. We also investigate the distribution of prime pairs, in connection which we offer original ideas. The Riemann zeta function is defined as ζ(s) := \sum_{n =1}^{∞} n^{-s} in the half-plane Re s > 1. We extend it to a meromorphic function on the whole plane with a simple pole at s=1, and show that it satisfies the functional equation. We discuss two methods, van der Corput's and Vinogradov's, to give upper bounds for the growth of the zeta function on the critical strip 0 ≤ Re s ≤ 1. Both of these are based on the observation that ζ(s) is well approximated on the critical strip by a finite exponential sum \sum_{n =1}^{T} n^{-s} = \sum_{n =1}^{T} exp\{ -s log n \}. Van der Corput's method uses the Poisson summation formula to transform this sum into a sum of integrals, which can be easily estimated. This yields the estimate ζ(1/2 + it) = \mathcal{O} (t^{\frac{1}{6}} log t), as t → ∞. Vinogradov's method transforms the problem of estimating an exponential sum into a combinatorial problem. It is needed to give a strong bound for the growth of the zeta function near the vertical line Re s = 1. We use complex analysis to prove the Hoheisel-Ingham Theorem, which states that if ζ(1/2 + it) = \mathcal{O} (t^{c}) for some constant c > 0, then for any θ > \frac{1+4c}{2+4c}, and for any function x^{θ} << h(x) << x, we have ψ (x+h) - ψ (x) ∼ h, as x → ∞. The proof of this relies heavily on the growth estimate obtained by the Vinogradov's method. Here ψ(x) := \sum_{n ≤ x} Λ (n) = \sum_{p^k ≤ x} log p is the summatory function of the von Mangoldt's function. From this we obtain by using van der Corput's estimate that the difference between consecutive primes satisfies p_{n+1} - p_{n} < p_{n}^{\frac{5}{8} + \epsilon} for all large enough n, and for any \epsilon > 0. Finally, we study prime pairs, and the Hardy-Littlewood Conjecture on their distribution. More precisely, let π _{2k}(x) stand for the number of prime numbers p ≤ x such that p+2k is also a prime. The following ideas are all original contributions of this thesis: We show that the average of π _{2k}(x) over 2k ≤ x^{θ} is exactly what is expected by the Hardy-Littlewood Conjecture. Here we can choose θ > \frac{1+4c}{2+4c} as above. We also give a lower bound of π _{2k}(x) for the averages over much smaller intervals 2k ≤ E log x, and give interpretations of our results using the concept of equidistribution. In addition, we study prime pairs by using the discrete Fourier transform. We express the function π _{2k}(n) as an exponential sum, and extract from this sum the term predicted by the Hardy-Littlewood Conjecture. This is interpreted as a discrete analog of the method of major and minor arcs, which is often used to tackle problems of additive number theory.