Power Saving Error Term Research Articles

Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata

Abstract We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

The twisted second moment of modular half-integral weight $L$-functions

Given a half-integral weight holomorphic Kohnen newform f on \Gamma_{0}(4) , we prove an asymptotic formula for large primes p with power saving error term for \sum_{\chi \pmod{p}}^{\qquad{}{}_{\scriptsize{*}}}| L(1/2,f,\chi) |^{2}. Our result is unconditional, it does not rely on the Ramanujan–Petersson conjecture for the form f . This gives a very sharp Lindelöf-on-average result for Dirichlet series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution problem involving the Fourier coefficients of half-integral weight forms. The second input is a bound for sums of products of Salié sums in the Pólya–Vinogradov range. Half-integrality is fully exploited to establish such an estimate. We use the closed form evaluation of the Salié sum to relate our problem to the sequence \alpha n^{2} \pmod{1} . Our treatment of this sequence is inspired by work of Rudnick–Sarnak and the second author on the local spacings of \alpha n^{2} modulo 1.

Rational points on a class of cubic hypersurfaces

Abstract Let r ⩾ 3 r\geqslant 3 be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 .

A supplement to Chebotarev’s density theorem

Let L/K be a Galois extension of number fields with Galois group G. We show that if the density of prime ideals in K that split totally in L tends to 1/∣G∣ with a power saving error term, then the density of prime ideals in K whose Frobenius is a given conjugacy class C ⊂ G tends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of ζL(s)/ζK(s).

Mean value of real Dirichlet characters using a double Dirichlet series

AbstractWe study the double character sum $\sum \limits _{\substack {m\leq X,\\m\mathrm {\ odd}}}\sum \limits _{\substack {n\leq Y,\\n\mathrm {\ odd}}}\left (\frac {m}{n}\right )$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages—it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.

The Second Moment Theory of Families of 𝐿-Functions–The Case of Twisted Hecke 𝐿-Functions

For a fairly general family of L L -functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q q , and prove that it satisfies such formulas. We derive arithmetic consequences: a positive proportion of central values L ( f ⊗ χ , 1 / 2 ) L(f\otimes \chi ,1/2) are non-zero, and indeed bounded from below; there exist many characters χ \chi for which the central L L -value is very large; the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.

Limit laws for rational continued fractions and value distribution of quantum modular forms

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in (0, 1] ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We obtain the Gaussian behaviour of central values of the Esterman function ∑ n ⩾ 1 τ ( n ) e 2 π i n x / n $\sum _{n\geqslant 1} \tau (n) {\rm e}^{2\pi i n x}/\sqrt {n}$ ( x ∈ Q $x\in {\mathbb {Q}}$ ), a problem for which known approaches based on Eisenstein series have been so far ineffective. We give a new proof, based on dynamical systems, that central modular symbols associated with a holomorphic cusp form for S L ( 2 , Z ) $SL(2,{\mathbb {Z}})$ have a Gaussian distribution, and give the first proof of an estimate for their probabilities of large deviations. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods.

Effective counting of simple closed geodesics on hyperbolic surfaces

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most L on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.

Uniform bounds for lattice point counting and partial sums of zeta functions

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice \(\Lambda \subseteq \mathbb {R}^d\) inside the d-sphere of radius R. In contrast to previous works, we obtain error terms with implied constants depending only on d. Secondly, let \(\phi (s) = \sum _n a(n) n^{-s}\) be a ‘well behaved’ zeta function. A classical method of Landau yields asymptotics for the partial sums \(\sum _{n < X} a(n)\), with power saving error terms. Following an exposition due to Chandrasekharan and Narasimhan, we obtain a version where the implied constants in the error term will depend only on the ‘shape of the functional equation’, implying uniform results for families of zeta functions with the same functional equation.

THE ASYMPTOTIC DISTRIBUTION OF TRACES OF WEAK MAASS FORMS

We give an asymptotic formula with a power saving error term for traces of CM values of a generic class of weak Maass forms. We apply this result to study the asymptotic distribution of several arithmetic functions, including the partition function, the Andrews smallest parts function, and the coefficients of Ramanujan's mock theta function f ( q ) .

Generalized divisor functions in arithmetic progressions: I

We prove some distribution results for the k-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length X of the sum, with appropriate constrains and averaging on the moduli, saving a power of X from the trivial bound. On assuming the Generalized Riemann Hypothesis, we obtain uniform power saving error terms that are independent of k.We follow and specialize Y.T. Zhang's method on bounded gaps between primes to our setting. Our arguments are essentially self-contained, with the exception on the use of Deligne's work on the Riemann Hypothesis for varieties over finite fields. In particular, we avoid the reliance on Siegel's theorem, leading to some effective estimates.

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Abstract Let $\varepsilon&gt; 0$ be sufficiently small and let $0 &lt; \eta &lt; 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

A Refined Conjecture for the Variance of Gaussian Primes across Sectors

We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges upon taking into account lower order terms. For sufficiently small sectors, we moreover prove an unconditional result that is consistent with our conjecture down to lower order terms.

Asymptotic trace formula for the Hecke operators

Given integers m, n and k, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the mth and nth Fourier coefficients of an orthonormal basis of $$S_k\left( N\right) ^*$$ (the weight k newforms with fixed square-free level N) provided that $$|4 \pi \sqrt{mn}- k|=o\left( k^{\frac{1}{3}}\right) $$ . Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator $$\mathcal {T}_n^*$$ on $$S_k\left( N\right) ^*$$ averaged over k in a short interval. By bounding the second moment of the trace of $$\mathcal {T}_{n}$$ over a larger interval, we show that the trace of $$\mathcal {T}_n$$ is unusually large in the range $$|4 \pi \sqrt{n}- k| = o\left( n^{\frac{1}{6}}\right) $$ . As an application, for any fixed prime p coprime to N, we show that there exists a sequence $$\{k_n\}$$ of weights such that the error term of Weyl’s law for $$\mathcal {T}_p$$ is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd et al. (J Eur Math Soc 1(1):51–85, 1999) [Theorem 1.4] with an improved exponent.

The asymptotic formula for the averages of the product of twisted L-functions

Suppose that q and r are two distinct large primes. Let g be a cuspidal Hecke eigenform of level 1 and even weight \(k_1\) and \(H_{k_2}(q)\) be the set of cuspidal Hecke eigenforms of level q and even weight \(k_2\). In this paper, we investigate the averages of the product of the central values of two L-functions of modular forms \(f\in H_{k_2}(q)\) and g twisted by all primitive Dirichlet characters modulo r and obtain an asymptotic formula with a power saving error term under certain assumptions.

On a certain non-split cubic surface

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for this surface.

High moments of the Estermann function

For $a/q\in\mathbb{Q}$ the Estermann function is defined as $D(s,a/q):=\sum_{n\geq1}d(n)n^{-s}\operatorname{e}(n\frac aq)$ if $\Re(s)>1$ and by meromorphic continuation otherwise. For $q$ prime, we compute the moments of $D(s,a/q)$ at the central point $s=1/2$, when averaging over $1\leq a<q$. As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions $\sum_{\chi_1,\dots,\chi_k\mod q}|L(\frac12,\chi_1)|^2\cdots |L(\frac12,\chi_k)|^2|L(\frac12,\chi_1\cdots \chi_k)|^2$, obtaining a power saving error term. Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing $f_{\pm}(a/q):=\sum_{j=0}^r (\pm1)^jb_j$ where $[0;b_0,\dots,b_r]$ is the continued fraction expansion of $a/q$ we prove that for $k\geq2$ and $q$ primes one has $\sum_{a=1}^{q-1}f_{\pm}(a/q)^k\sim2 \frac{\zeta(k)^2}{\zeta(2k)} q^k$ as $q\to\infty$.

Spectral summation formula for GSp(4) and moments of spinor $L$-functions

We compute the first and second moment of the spinor L -function at the central point of Siegel modular forms of large weight k with power saving error term and give applications to non-vanishing.

Linear correlations of the divisor function

Motivated by arithmetic applications on the number of points in a bihomogeneous variety and on moments of Dirichlet $L$-functions, we provide analytic continuation for the series $\mathcal A_{\boldsymbol{a}}(s):=\sum_{n_1,\dots,n_k\geq1}\frac{d(n_1)\cdots d(n_k)}{(n_1\cdots n_k)^{s}}$ with the sum restricted to solutions of a non-trivial linear equation $a_1n_1+\cdots+a_kn_k=0$. The series $\mathcal A_{\boldsymbol{a}}(s)$ converges absolutely for $\Re(s)>1-\frac1k$ and we show it can be meromorphically continued to $\Re(s)>1-\frac 2{k+1}$ with poles at $s=1-\frac1{k-j}$ only, for $1\leq j< (k-1)/2$. As an application, we obtain an asymptotic formula with power saving error term for the number of points in the variety $a_1x_1y_1+\cdots+a_kx_ky_k=0$ in $\mathbb P^{k-1}(\mathbb Q)\times \mathbb P^{k-1}(\mathbb Q)$.

Triple correlations of ternary divisor functions

Let d3(n) denote the ternary divisor function, and d(n) be the divisor function. In this article we shall be concerned with the triple correlation of the formS(a,d,d3;N,H):=∑hW(hH)∑N⩽n⩽2Na(n)d(n−h)d3(n+h), where W is a smooth function with compact support in [1,2]. We succeed in obtaining an asymptotic formula with a power-saving error term.