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Étienne Fouvry

Publications and source records attributed to Étienne Fouvry.

30 records · Page 2Linked to original sources
arXiv2018

Another application of Linnik's dispersion method

Let $α_m$ and $β_n$ be two sequences of real numbers supported on $[M, 2M]$ and $[N, 2N]$ with $M = X^{1/2 - δ}$ and $N = X^{1/2 + δ}$. We show that there exists a $δ_0 > 0$ such that the multiplicative convolution of $α_m$ and $β_n$ has exponent of distribution $\frac{1}{2} + δ-\varepsilon$ (in a weak sense) as long as $0 \leq δ< δ_0$, the sequence $β_n$ is Siegel-Walfisz and both sequences $α_m$ and $β_n$ are bounded above by divisor functions. Our result is thus a general dispersion estimate for "narrow" type-II sums. The proof relies crucially on Linnik's dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.

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arXiv2018

Level of distribution of unbalanced convolutions

We show that if an essentially arbitrary sequence supported on an interval containing $x$ integers, is convolved with a tiny Siegel-Walfisz-type sequence supported on an interval containing $\exp((\log x)^{\varepsilon})$ integers then the resulting multiplicative convolution has (in a weak sense) level of distribution $x^{1/2 + 1/66 - \varepsilon}$ as $x$ goes to infinity. This dispersion estimate has a number of consequences for: the distribution of the $k$th divisor function to moduli $x^{1/2 + 1/66 - \varepsilon}$ for any integer $k \geq 1$, the distribution of products of exactly two primes in arithmetic progressions to large moduli, the distribution of sieve weights of level $x^{1/2 + 1/66 - \varepsilon}$ to moduli as large as $x^{1 - \varepsilon}$ and for the Brun-Titchmarsh theorem for almost all moduli $q$ of size $x^{1 - \varepsilon}$, lowering the long-standing constant $4$ in that range. Our result improves and is inspired by earlier work of Green (and subsequent work of Granville-Shao) which is concerned with the distribution of $1$-bounded multiplicative functions in arithmetic progressions to large moduli. As in these previous works the main technical ingredient are the recent estimates of Bettin-Chandee for trilinear forms in Kloosterman fractions and the estimates of Duke-Friedlander-Iwaniec for bilinear forms in Kloosterman fractions.

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arXiv2016

Some applications of smooth bilinear forms with Kloosterman sums

We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.

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arXiv2016

On moments of twisted $L$-functions

We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ and $g$ is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted $L$-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M.~Young's asymptotic evaluation of the fourth moment of Dirichlet $L$-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.

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arXiv2014

Algebraic twists of modular forms and Hecke orbits

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the ell-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.

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arXiv2014

Algebraic trace functions over the primes

We study sums over primes of trace functions of $\ell$-adic sheaves. Using an extension of our earlier results on algebraic twist of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann Hypothesis over finite fields, we prove general estimates with power-saving for such sums. We then derive various concrete applications.

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arXiv2014

On the exponent of distribution of the ternary divisor function

We show that the exponent of distribution of the ternary divisor function $d_3$ in arithmetic progressions to prime moduli is at least 1/2+1/46, improving results of Heath-Brown and Friedlander--Iwaniec. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to 1/2 +1/34.

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arXiv2013

On the greatest prime factor of ab+1

We improve some results on the size of the greatest prime factor of integers of the form ab+1, where a and b belong to finite sets of integers with rather large density.

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arXiv2013

Counting sheaves using spherical codes

Using the Riemann Hypothesis over finite fields and bounds for the size of spherical codes, we give explicit upper bounds, of polynomial size with respect to the size of the field, for the number of geometric isomorphism classes of geometrically irreducible ell-adic middle-extension sheaves on a curve over a finite field which are pointwise pure of weight 0 and have bounded ramification and rank. As an application, we show that "random" functions defined on a finite field can not usually be approximated by short linear combinations of trace functions of sheaves with small complexity.

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arXiv2013

Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.

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