Power Saving Error Term Research Articles

On some multivariate LCM and GCD sums

In this paper we obtain an asymptotic formula with a power saving error term for the summation function of a family of generalized least common multiple and greatest common divisor functions of several integer variables.

SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to$P_{k}(n)^{2}$, where$P_{k}(n)$is the discrepancy between the volume of the$k$-dimensional sphere of radius$\sqrt{n}$and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including$\sum P_{k}(n)^{2}e^{-n/X}$and the Laplace transform$\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions$k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums$\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral$\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.

On ℓ-torsion in class groups of number fields

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-$D_4$) and quintic fields with chosen splitting types at a finite set of primes.

The Shifted Convolution of Generalized Divisor Functions

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously by Fouvry and Tenenbaum and, more recently, by 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 L-functions, we prove power-saving error term in the Titchmarsh divisor problem of estimating ∑ p ⩽ x τ ( p − 1 ) . 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 ∑ n ⩽ x τ k ( n ) τ ( n + 1 ) , reproving a result announced by Bykovskiĭ and Vinogradov by a different method. The gain in the exponent is shown to be independent of k if a generalized Lindelöf hypothesis is assumed.

On a certain additive divisor problem

We prove an asymptotic formula for a variant of the binary additive divisor problem with linear factors in the arguments, which has a power saving error term and which is uniform in all involved parameters.

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.

On triple correlations of divisor functions

We prove an asymptotic formula with power saving error term for a certain triple divisor sum.

THE SHIFTED CONVOLUTION OF DIVISOR FUNCTIONS

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_3(n)$ and $d(n)$, which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates for the shifted convolution of $d_3(n)$ and Fourier coefficents of holomorphic cusp forms. These asymptotics improve previous results obtained by several different authors.

Singular moduli and the distribution of partition ranks modulo 2

AbstractIn this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.

The asymptotic distribution of Andrews’ smallest parts function

In this paper, we use methods from the spectral theory of automorphic forms to give an asymptotic formula with a power saving error term for Andrews’ smallest parts function spt(n). We use this formula to deduce an asymptotic formula with a power saving error term for the number of 2-marked Durfee symbols associated to partitions of n. Our method requires that we count the number of Heegner points of discriminant −D < 0 and level N inside an “expanding” rectangle contained in a fundamental domain for \({\Gamma_0(N)}\).

The average of the divisor function over values of a quadratic polynomial

We establish a uniform asymptotic formula with a power saving error term for the average of the divisor function τ(n) := ∑ k|n 1 over values of the quadratic polynomial x 2 + |D| where D < 0 is a fundamental discriminant.

The Second Moment of Twisted Modular L-Functions

We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Applications are given to simultaneous non-vanishing and lower bounds on higher moments of twisted L-functions.

COUNTING -FIELDS WITH A POWER SAVING ERROR TERM

AbstractWe show how the Selberg $\Lambda ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using the geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.

On a certain senary cubic form

A strong form of the Manin-Peyre conjecture with a power saving error term is proved for a certain cubic fourfold.

Automorphic spectral identities and applications to automorphic L -functions onGL2

We prove an automorphic spectral identity on GL 2 involving second moments. From it we obtain an asymptotic, with power-saving error term, for (non-archimedean) conductor-aspect integral moments, twisting by GL 1 characters ramifying at a fixed finite place. The strength of the spectral identity, and of the resulting asymptotics, is illustrated by extracting a subconvex bound in conductor aspect at a fixed finite prime.

A hybrid asymptotic formula for the second moment of Rankin-Selberg L -functions

Let g be a fixed modular form of full level, and let {fj, k} be a basis of holomorphic cuspidal newforms of even weight k, fixed level and fixed primitive nebentypus. We consider the Rankin–Selberg L-functions L(½+it, fj, k ⊗ g) and compute their second moment over t ≈ T and k ≈ K. For K3/4+ε⩽ T ⩽ K5/4−ε, we obtain an asymptotic formula with a power-saving error term. Our result covers the second moment of L(½+it+ir, fj, k)L(½+it−ir, fj, k) for any fixed real number r, hence also the fourth moment of L(½+it, fj, k). For the proof, we develop a precise uniform approximate functional equation with explicit dependence on the archimedean parameters.