Short Sums Research Articles

Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

AbstractLet be a random matrix distributed according to uniform probability measure on the finite general linear group . We show that equidistributes on as as long as and that this range is sharp. We also show that nontrivial linear combinations of equidistribute as long as and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for , where depends on , due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of , we end up showing that certain explicit character sums modulo exhibit cancellation when averaged over monic polynomials of degree in as long as . This goes far beyond the classical range due to Montgomery and Vaughan. To study these sums, we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for [Formula: see text]. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for [Formula: see text]. Furthermore, we present a conditional result regarding sign changes of these coefficients.

Large sums of high‐order characters

AbstractLet be a primitive character modulo a prime , and let . It has previously been observed that if has large order then for some , in analogy with Vinogradov's conjecture on quadratic non‐residues. We give a new and simple proof of this fact. We show, furthermore, that if is squarefree then for any th root of unity the number of such that is whenever . Consequently, when has sufficiently large order the sequence cannot cluster near for any . Our proof relies on a second moment estimate for short sums of the characters , averaged over , that is non‐trivial whenever has no small prime factors. In particular, given any we show that for all but powers , the partial sums of exhibit cancellation in intervals as long as is prime, going beyond Burgess' theorem. Our argument blends together results from pretentious number theory and additive combinatorics. Finally, we show that, uniformly over prime , the Pólya–Vinogradov inequality may be improved for on average over , extending work of Granville and Soundararajan.

Three conjectures about character sums

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L(1,chi ) and L(1+it,chi )) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

Mucomucosal anastomotic non-transecting augmentation (MANTA) urethroplasty: a ventral modification for obliterative strictures.

To present a surgical modification for the repair of bulbar urethral strictures containing short, highly obliterative segments and report on long-term objective and patient-reported outcomes. We considered patients undergoing bulbar buccal mucosal graft urethroplasty (BMGU) between July 2016 and December 2019. Eligibility criteria for mucomucosal anastomotic non-transecting augmentation (MANTA) urethroplasty were strictures of ≥2 cm with an obliterative segment of ≤1.5 cm. The stricture is approached ventrally to avoid extensive dissection and mobilisation. Dorsally, the scar is superficially excised and the spongiosum is left intact. Dorsal mucomucosal anastomosis is complemented by ventral onlay graft. Perioperative characteristics were prospectively collected including uroflowmetry data and validated patient-reported outcome measures on voiding, erectile, and continence function. We evaluated functional follow-up, incorporating patient-reported (lower urinary tract symptoms [LUTS] score) and functional success. Recurrence was defined as need of re-treatment. Of 641 men treated with anterior BMGU, 54 (8.4%) underwent MANTA urethroplasty. Overall, 26 (48%) and 45 (83%) had a history of dilatation and urethrotomy, respectively, and 14 (26%) were redo cases. Location was bulbar in 38 (70%) and penobulbar in 16 patients (30%), and the mean (SD) graft length was 4.5 (1.4) cm. At a median (interquartile range) follow-up of 41 (27-53) months, the functional success rate was 93%. Whereas the median LUTS score significantly improved from baseline to postoperatively (13 vs 3.5; P < 0.001), there was no change in erectile function (median International Index of Erectile Function - erectile function domain score 27 vs 24) or urinary continence (median International Consultation on Incontinence Questionnaire - Urinary Incontinence Short Form sum score 0 vs 0; all P ≥ 0.4). All patients were 'satisfied' (27%) or 'very satisfied' (73%) with the outcome of their operation. With excellent long-term objective and patient-reported outcomes, MANTA urethroplasty adds to the armamentarium for long bulbar strictures with a short obliterative segment.

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

Weyl-type bounds for twisted GL(2) short character sums

Let f be a Hecke–Maass or holomorphic primitive cusp form of full level for \(SL(2,{\mathbb {Z}})\) with normalized Fourier coefficients \(\lambda _{f}(n)\). Let \(\chi \) be a primitive Dirichlet character of modulus p, a prime. In this article, we shorten the range of cancellation for N in the twisted GL(2) short character sum. Here, we consider the problem of cancellation in short character sum of the form $$\begin{aligned} S_{f,\chi }(N):= \mathop \sum _{n \in {\mathbb {Z}}}\lambda _{f}(n)\chi (n)W\Big (\frac{n}{N}\Big ). \end{aligned}$$We show that, for \(0<\theta < \frac{1}{10}\), $$\begin{aligned} S_{f,\chi }(N) \ll _{f,\epsilon }N^{3/4 + \theta /2}p^{1/6}(pN)^{\epsilon } + N^{1-\theta }(pN)^{\epsilon }, \end{aligned}$$which is non-trivial if \(N \ge p^{2/3 + \alpha + \epsilon }\) where\(\alpha = = \frac{4\theta }{1-6\theta }\). Previously, such a bound was known for \(N \ge p^{3/4 + \epsilon }.\)

Möbius cancellation on polynomial sequences and the quadratic Bateman–Horn conjecture over function fields

We establish cancellation in short sums of certain special trace functions over \({\mathbb {F}}_q[u]\) below the Pólya–Vinogradov range, with savings approaching square-root cancellation as q grows. This is used to resolve the \({\mathbb {F}}_q[u]\)-analog of Chowla’s conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman–Horn conjecture in degree 2, for some values of q. A final application is to sums of trace functions over primes in \({\mathbb {F}}_q[u]\).

Sommes courtes de certaines fonctions multiplicatives

The purpose of this paper is to establish upper bounds for some short sums of a class of multiplicative functions over integers with certain restrictions on the number of prime factors. More precisely, we consider sums expressed in the form [Formula: see text]; or more generally [Formula: see text] where f is a positive multiplicative function, G is a polynomial with integer coefficients, and as usual, [Formula: see text] denotes the number of distinct prime factors of the integer m.

From ESPRIT to ESPIRA: estimation of signal parameters by iterative rational approximation

AbstractWe introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method (MPM) applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices that are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the MPM is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the MPM, but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the MPM for noisy data and for signal approximation by short exponential sums.

Vinogradov’s sieve and an estimate for an incomplete Kloosterman sum

We refine a bound for a short Kloosterman sum with a prime modulus using the so-called Vinogradov sieve. The number of terms in the sum can be less than an arbitrarily small fixed power of . Bibliography: 26 titles.

Arithmetic exponent pairs for algebraic trace functions and applications

We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As an application, we prove a quadratic analogue of Brun-Titchmarsh theorem on average, bounding the number of primes $p\leqslant X$ with $p^2+1\equiv0\pmod q$. The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by Irving.

A Density of Ramified Primes

Let K be a cyclic number field of odd degree over {mathbb {Q}} with odd narrow class number, such that 2 is inert in K/{mathbb {Q}}. We define a family of number fields {K(p)}_p, depending on K and indexed by the rational primes p that split completely in K/{mathbb {Q}}, in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in K(p)/{mathbb {Q}} is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension K/{mathbb {Q}}. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

Monte Carlo MP2-F12 for Noncovalent Interactions: The C60 Dimer.

A scalable stochastic algorithm is presented that can evaluate explicitly correlated (F12) second-order many-body perturbation (MP2) energies of weak, noncovalent, intermolecular interactions. It first transforms the formulas of the MP2 and F12 energy differences into a short sum of high-dimensional integrals of Green's functions in real space and imaginary time. These integrals are then evaluated by the Monte Carlo method augmented by parallel execution, redundant-walker convergence acceleration, direct-sampling autocorrelation elimination, and control-variate error reduction. By sharing electron-pair walkers across the supermolecule and its subsystems spanned by the joint basis set, the statistical uncertainty is reduced by one to 2 orders of magnitude in the MP2 binding energy corrected for the basis-set incompleteness and superposition errors. The method predicts the MP2-F12/aug-cc-pVDZ binding energy of 19.1 ± 4.0 kcal mol-1 for the C60 dimer at the center distance of 9.748 Å.

Cancellation in algebraic twisted sums on GL_ m

Abstract Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε &gt; 0 {\varepsilon&gt;0} .

Joint distribution of spins

We answer a question of Friedlander, Iwaniec, Mazur, and Rubin on the joint distribution of spin symbols. As an application we give a negative answer to a conjecture of Cohn and Lagarias on the existence of governing fields for the 16-rank of class groups under the assumption of a short character sum conjecture.

An investigation into explicit versions of Burgess' bound

Let χ be a Dirichlet character modulo p, a prime. In applications, one often needs estimates for short sums involving χ. One such estimate is the family of bounds known as Burgess' bound. In this paper, we explore several adjustments one can make to the work of Treviño [11] in obtaining explicit versions of Burgess' bound. For an application, we investigate the problem of the existence of a kth power non-residue modulo p which is less than pα for several fixed α. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields found in [7].

On Short Sums Involving Fourier Coefficients of Maass Forms

We study sums of Hecke eigenvalues of Hecke-Maass cusp forms for the group $\mathrm{SL}(n,\mathbb Z)$, with general $n\geq 3$, over certain short intervals under the assumption of the generalised Lindelof hypothesis and a slightly stronger upper bound concerning the exponent towards the Ramanujan-Petersson conjecture that is currently known. In particular, in this case we evaluate the second moment of the sums in question asymptotically.

Estimates of short exponential sums with primes in major arcs

Estimates of short exponential sums with primes in major arcs

Short Character Sums and the Pólya–Vinogradov Inequality

Abstract We show in a quantitative way that any odd primitive character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to $$\begin{equation*} \max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q) \end{equation*}$$ then for any ɛ &amp;gt; 0 one may exhibit cancellation in partial sums of χ on the interval [1, t] whenever $t \gt q^{\varepsilon}$, i.e., $$\begin{equation*} \sum_{n \leq t} \chi(n) = o_{q \rightarrow \infty}(t)\ \text{for all } t \gt q^{\varepsilon}. \end{equation*}$$ We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya–Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.