Large Sieve Research Articles

A larger GL 2 large sieve in the level aspect

Abstract In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL 2 cusp forms modular with respect to the congruence subgroups Γ1(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q 2−δ for any δ > 0, where N is the length of linear forms in the large sieve.

Small gaps between zeros of twisted L-functions

We use the asymptotic large sieve, developed by the authors, to prove that if the Generalized Riemann Hypothesis is true, then there exist many Dirichlet L-functions that have a pair of consecutive zeros closer together than 0.37 times their average spacing. More generally, we investigate zero spacings within the family of twists by Dirichlet characters of a fixed L-function and give precise bounds for small gaps which depend only on the degree of the L-function.

PERFORMANCE OF Ce-INCORPORATED KIT-6 SUPPORTED COBALT CATALYSTS FOR FISCHER–TROPSCH SYNTHESIS

A series of Ce -incorporated structure of cubic large mesoporous molecular sieves, KIT-6, with different Ce contents were synthesized by a direct hydrothermal process. A sample of Ce -loaded KIT-6 material was synthesised by incipent wetness impregnation. Similarly, catalysts with 15 wt.% Co loading on the above supports were also synthesized by incipient wetness impregnation. The supports and catalysts were characterized by X-ray diffraction, diffuse reflectance UV-vis, solid-state 29Si magic-angle spinning nuclear magnetic resonance, H2 -temperature programmed reduction, H2 -temperature programmed desorption and oxygen titration. The structure of the KIT-6 support was well retained after Ce incorporation. Small amounts of Ce in the Co catalyst were found to improve the activity and increase the selectivity to C5+ hydrocarbons for Fischer–Tropsch synthesis, while larger amounts of Ce had the reverse effect. Meanwhile, methane selectivity shows an opposite trend as compared with that of C5+ selectivity. Ce -loaded KIT-6 supported Co catalyst showed lower activity than KIT-6 supported Co catalyst.

Fermat quotients: exponential sums, value set and primitive roots

For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with $N$ consecutive Fermat quotients that is nontrivial for $N\ge p^{1/2+\epsilon}$ for any fixed $\epsilon>0$. We use a recent idea of M. Z. Garaev together with a form of the large sieve inequality due to S. Baier and L. Zhao, to show that on average over $p$ one can obtain a nontrivial estimate for much shorter sums starting with $N\ge p^{\epsilon}$. We also obtain lower bounds on the image size of the first $N$ consecutive Fermat quotients and use it to prove that there is a positive integer $n\le p^{3/4 + o(1)}$ such that $q_p(n)$ is a primitive root modulo $p$.

LARGE SIEVE INEQUALITIES FOR QUARTIC CHARACTERS

In this paper, we prove a large sieve inequality for quartic Dirichlet characters. The result is analogous to large sieve inequalities for the quadratic and cubic Dirichlet characters.

Selection of an ideal mesh size for the cracking unit of a palm kernel processing plant

One of the main problems associated with cracking of palm nut is the mixture of small Tenera and the big Dura variety nuts. In general, low yield of about 40% is obtained from cracking the mixture of Dura-Tenera nuts. In cracking the mixture, most of the Tenera nuts are not cracked when a large sieve size is used. When the sieve size is small, the Dura kernels get broken, thereby affecting the quality of the processed kernel oil. Hence the objective of this work was to develop a model for selecting a suitable sieve size that can be used to crack the nut in order to increase productivity and also improve the quality of the palm kernel oil. Series of experiments carried out confirmed the need for separation of nuts before cracking and also the need to select an ideal sieve size for each type of nut. In conclusion, it was established that separating the two nut varieties before cracking led to a significant (p< 0.01) increase in cracking efficiency of up to about 90% thereby increasing the productivity by 40% and making an economic gain of GH¢ 1.07 (US$0.98, October, 2008) per every 50 kg nut processed.

Applications of the asymptotic large sieve

Applications of the asymptotic large sieve

Splitting fields of elements in arithmetic groups

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

&lt;i&gt;Echinoderes rex&lt;/i&gt; n. sp. (Kinorhyncha: Cyclorhagida), the largest &lt;i&gt;Echinoderes&lt;/i&gt; species found so far

A new kinorhynch species, Echinoderes rex n. sp., is described from the Korea Strait. The new species is characterized by a pair of diminutive lateral terminal spines (19-23 μm) and a trunk length of 482-528 μm, making it the largest Echinoderes described so far. Unique for the new species is also the presence of a type of putative glandular cell outlets that have not been described previously. This paper presents light and scanning electron micrographs of the new structure. In addition, E. rex n. sp. is characterized by having a single middorsal spine on segment 4, lateroventral tubules on segment 5 and 8 and lateroventral acicular spines on segments 6 and 7. A pair of distinct and conspicuously large sieve plates, only described from two other species of the genus, is present on segment 9 in E. rex n. sp. A comparison is made with other species of Echinoderes and the similarities are discussed.

Enlarged major arcs in additive problems

Enlarged major arcs in the Waring-Goldbach problem are studied by using large sieve estimates for Dirichlet polynomials and estimates for exponential sums over primes.

The Large Sieve, Monodromy, and Zeta Functions of Algebraic Curves, 2: Independence of the Zeros

Using the sieve for Frobenius developed earlier by the author, we show that in a certain sense, the roots of the L-functions of most algebraic curves over finite fields do not satisfy any nontrivial (linear or multiplicative) dependency relations. This can be seen as an analogue of conjectures of Q-linear independence among ordinates of zeros of L-functions over number fields. As a corollary of independent interest, we find for “most” pairs of distinct algebraic curves over a finite field the form of the distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The method of proof also emphasizes the relevance of random matrix models for this type of arithmetic questions. We also describe an alternate approach, suggested by Katz, which relies on Serre's theory of Frobenius tori.

A Large Sieve Inequality of Elliott-Montgomery-Vaughan Type for Automorphic Forms and Two Applications

In this paper, we establish a large sieve inequality of Elliott-Montgomery-Vaughan type for Fourier coefficients of newforms. As applications, we give a significant improvement on the principal result of Duke and Kowalski on Linnik's problem for modular forms and prove the upper part of the first conjecture of Montgomery-Vaughan in the context of automorphic L-functions.

A new bound K2/3+ for Rankin-Selberg L-functions for Hecke congruence subgroups

Let f be a holomorphic Hecke eigenform for Γ0(N ) of weight k, or a Maass eigenform for Γ0(N ) with Laplace eigenvalue 1/4 + k. Let g be a fixed holomorphic or Maass cusp form for Γ0(N ). A subconvexity bound for central values of the Rankin-Selberg L-function L(s, f ⊗ g) is proved in the k-aspect: L(1/2+ it, f ⊗ g) N ,g,t,e k, while a convexity bound is only k. This new bound improves earlier subconvexity bounds for these Rankin-Selberg L-functions by Sarnak, the authors, and Blomer. Techniques used include a result of Good, spectral large sieve, meromorphic continuation of a shifted convolution sum to −1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.

The large sieve and random walks on left cosets of arithmetic groups

Building on Kowalski’s work on random walks on the groups \mathrm{SL}(n,ℤ) and \mathrm{Sp}(2g,ℤ) , we consider similar problems (we try to estimate the probability with which, after k steps, the matrix obtained has a characteristic polynomial with maximal Galois group or has no nonzero squares among its entries) for more general classes of sets: in \mathrm{GL}(n,A) , where A is a subring of ℚ containing ℤ that we specify, we perform a random walk on the set of matrices with fixed determinant D \in A^× . We also investigate the case where the set involved is any of the two left cosets of the special orthogonal group \mathrm{SO}(n,m)(ℤ) with respect to the spinorial kernel Ω(n,m)(ℤ) .

Amplification arguments for large sieve inequalities

We give a new proof of the arithmetic large sieve inequality based on an amplification argument, and use a similar method to prove a new sieve inequality for classical holomorphic cusp forms. A sample application of the latter is also given.

Eigenvalues in the large sieve inequality, II

Nous explorons numeriquement les valeurs de la forme hermitienne ∑ ∑ |∑ ϕ n e(na/q)| 2 q≤Q a mod * q n≤N lorsque N = ∑ q≤Q φ(q). Nous ameliorons la majoration actuelle et exhibons un graphique conjectural de la distribution asymptotique de ses valeurs propres en exploitant des resultats de calculs intensifs. L'une des consequences est que la distribution asymptotique existe probablement mais n'est pas absolument continue par rapport a la mesure de Lebesgue.

Sieving for mass equidistribution

We approach the holomorphic analogue to the Quantum Unique Ergodicity conjecture through an application of the Large Sieve. We deal with shifted convolution sums as in [Ho109], with various simplifications in our analysis due to the knowledge of the Ramanujan-Petersson conjecture in this holomorphic case.

Almost all elliptic curves are Serre curves

Using a multidimensional large sieve inequality, we obtain a bound for the mean-square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.

A generalization of the Barban–Davenport–Halberstam Theorem to number fields

For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q ⩽ Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the Barban–Davenport–Halberstam Theorem.

Zero-density estimates for automorphic L-functions

In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L(s, f ⊗ χ), where f is a holomorphic cusp form and χ(mod q) is a primitive character.