Olivier Ramaré

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli.

Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients

The distribution of squarefree binomial coefficients . For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they appear. It is evident that, if k is not too small, then must be highly composite in that it contains many prime factors and often to high powers. It is therefore of interest to enquire as to how infrequently is squarefree. One well-known conjecture, due to Erdős, is that is not squarefree once n > 4. Sarkozy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

Short effective intervals containing primes

Abstract We prove that every interval ]x(1−Δ−1),x] contains a prime number with Δ=28 314 000 and provided x⩾10 726 905 041 . The proof combines analytical, sieve and algorithmical methods.

An explicit density estimate for Dirichlet -series

Dirichlet L-series L(s, χ) = ∑ n≥1 χ(n)n −s associated to primitive Dirichlet characters χ are one of the keys to the distribution of primes. Even the simple case χ = 1 which corresponds to the Riemann zeta-function contains many informations on primes and on the Farey dissection. There have been many generalizations of these notions, and they all have arithmetical properties and/or applications, see [45, 29, 33] for instance. Investigations concerning these functions range over many directions, see [14] or [43]. We note furthermore that Dirichlet characters have been the subject of numerous studies, see [2, 50, 4]; Dirichlet series in themselves are still mysterious, see [3] and [6]. One of the main problem concerns the location of the zeroes of these functions in the strip 0 < <s < 1; the Generalized Riemann Hypothesis asserts that all of those are on the line <s = 1/2. We concentrate in this paper on estimating

Explicit estimates on several summatory functions involving the Moebius function

We prove that | ∑ d≤x μ(d)/d| log x ≤ 1/69 when x ≥ 96 955 and deduce from that: ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d)/d ∣ ∣ ∣ ∣ log(x/q) ≤ 4 5 q/φ(q) for every x > q ≥ 1. We also give better constants when x/q is larger. Furthermore we prove that |1 − ∑ d≤x μ(d) log(x/d)/d| ≤ 3 14 / log x and several similar bounds, from which we also prove corresponding bounds when summing the same quantity, but with the additional condition (d, q) = 1. We prove similar results for ∑ d≤x μ(d) log (x/d)/d, among which we mention the bound | ∑ d≤x μ(d) log (x/d)/d − 2 log x + 2γ0| ≤ 5 24/ log x, where γ0 is the Euler constant. We complete this collection by bounds such as ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d) ∣ ∣ ∣ ∣/x ≤ q φ(q) / log(x/q). We also provide all these bounds with variations where 1/ log x is replaced by 1/(1 + log x).

Explicit upper bounds for the remainder term in the divisor problem

We first report on computations made using the GP/PARI package that show that the error term ∆(x) in the divisor problem is

Arithmetical aspects of the large sieve inequality

= \mathscr{M} (x, 4) + O^* (0.35 x^{1/4} \log x)

A geometrical interpretation

when

On sums of seven cubes

x

The Siegel zero effect

ranges