Sary Drappeau

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the smooth summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet

Théorèmes de type Fouvry–Iwaniec pour les entiers friables

L

Sur les solutions friables de l'équation a+b=c

-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating

SMOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN ARITHMETIC PROGRESSIONS BEYOND THE

sum_{pleq x}tau(p-1)

x^{1/2}

. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for

-BARRIER

sum_{nleq x}tau_k(n)tau(n+1)

Sommes friables d'exponentielles et applications

, reproving a result announced by Bykovskiu{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of

On the average distribution of divisors of friable numbers

k

Weyl sums, mean value estimates, and Waring's problem with friable numbers

if a generalized Lindelof hypothesis is assumed.

Propriétés multiplicatives des entiers friables translatés

An integer n is said to be y-friable if its largest prime factor