* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Context The concept of citizen science (CS) is currently referred to by many actors inside and outside science and research. Several descriptions of this purportedly new approach of science are often heard in connection with large datasets and the possibilities of mobilizing crowds outside science to assists with observations and classifications. However, other accounts refer to CS as a way of democratizing science, aiding concerned communities in creating data to influence policy and as a way of promoting political decision processes involving environment and health. Objective In this study we analyse two datasets (N = 1935, N = 633) retrieved from the Web of Science (WoS) with the aim of giving a scientometric description of what the concept of CS entails. We account for its development over time, and what strands of research that has adopted CS and give an assessment of what scientific output has been achieved in CS-related projects. To attain this, scientometric methods have been combined with qualitative approaches to render more precise search terms. Results Results indicate that there are three main focal points of CS. The largest is composed of research on biology, conservation and ecology, and utilizes CS mainly as a methodology of collecting and classifying data. A second strand of research has emerged through geographic information research, where citizens participate in the collection of geographic data. Thirdly, there is a line of research relating to the social sciences and epidemiology, which studies and facilitates public participation in relation to environmental issues and health. In terms of scientific output, the largest body of articles are to be found in biology and conservation research. In absolute numbers, the amount of publications generated by CS is low (N = 1935), but over the past decade a new and very productive line of CS based on digital platforms has emerged for the collection and classification of data.

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What Is Citizen Science?--A Scientometric Meta-Analysis.

Consider the process D

 k
 , k = 1,2,…, given by
 
B

 i
 being independent standard Brownian motions. This process describes the limiting behavior “near the edge” in queues in series, totally asymmetric exclusion processes or oriented percolation. The problem of finding the distribution of D. was posed in [GW]. The main result of this paper is that the process D. has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from Gaussian Unitary Ensemble.

GUEs and queues

For any m≥1, let H
 m
 denote the quantity . A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for H
 m
 for m≥2, and specifically that H
 m
 ≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of Goldston, Pintz, and Yildirim, as well as the bound H
 m
 ≪m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1≤246 unconditionally and H1≤6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h1,n+h2,n+h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H1≤6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound or H
 m
 ≪m e2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H
 m
 when m=2,3,4,5.

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Variants of the Selberg sieve, and bounded intervals containing many primes

We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment 

$$\underset{\chi \bmod {q}}{\left. \sum \right.^{\ast}} L(1/2, f_1\otimes \chi) \overline{L(1/2, f_2 \otimes \chi)}$$

of central values of L-functions of any two (possibly equal) fixed cusp forms f1, f2 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 a new large sieve type bound for Kloosterman sums where the summation lengths can be 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.

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The Second Moment of Twisted Modular L-Functions

We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese remainder theorem conditions, obtaining an exponent of distribution 1/2 + 7/300.

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New equidistribution estimates of Zhang type

Reference EPFL-ARTICLE-202508doi:10.1090/S0894-0347-2013-00779-1View record in Web of Science Record created on 2014-10-23, modified on 2017-12-10

http://math.ou.edu/~apitale/research/jams779.pdf

Bounds for Rankin-Selberg integrals and quantum unique ergodicity for powerful levels

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing square-free level q -> infinity. We prove that the pushforward of the mass off to the modular curve of level 1 equidistributes with respect to the Poincare measure.

Equidistribution Of Cusp Forms In The Level Aspect

Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. 
We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions of Watson's formula to certain triple product integrals involving forms of non-squarefree level. By a theorem of Ichino and a lemma of Michel--Venkatesh, this amounts to a detailed study of Rankin--Selberg integrals int|f|^2 E attached to newforms f of arbitrary level and Eisenstein series E of full level. 
We find that the local factors of such integrals participate in many amusing analogies with global L-functions. For instance, we observe that the mass equidistribution conjecture with a power savings in the depth aspect is equivalent to the union of a global subconvexity bound and what we call a "local subconvexity bound"; a consequence of our local calculations is what we call a "local Lindelof hypothesis".

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincar\'{e} measure 
Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel and Vanderkam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994) 
Our proof follows the strategy of Holowinsky and Soundararajan (2008) who show that newforms of level 1 and large weight have equidistributed mass The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky's reduction of the problem to one of bounding shifted sums of Fourier coefficients (which on the surface makes sense only in the large weight limit), an evaluation of the p-adic integral needed to extend Watson's formula to the case of three newforms where the level of one divides but need not equal the common squarefree level of the other two, and some additional technical work in the problematic case that the level has many small prime factors

Paul D. Nelson

Papers

New equidistribution estimates of Zhang type

Bounds for Rankin-Selberg integrals and quantum unique ergodicity for powerful levels

Equidistribution Of Cusp Forms In The Level Aspect

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Equidistribution of cusp forms in the level aspect