Kannan Soundararajan

Bio: Kannan Soundararajan is an academic researcher from Stanford University. The author has contributed to research in topics: Riemann hypothesis & Number theory. The author has an hindex of 38, co-authored 174 publications receiving 4564 citations. Previous affiliations of Kannan Soundararajan include Princeton University & American Institute of Mathematics.

Demography of Cirsium vulgare in a grazing experiment.

Abstract: The complete demographies of 16 populations of Cirsium vulgare were followed in a replicated experiment. The experiment was a factorial combination of two intensities of sheep grazing in each of three seasons - winter (grazed or ungrazed), spring (grazed or ungrazed), and summer (light or heavy grazing) - giving eight treatments in two blocks. For 6 years from 1987 to 1992 the population sizes of C. vulgare were monitored in each of the 16 paddocks. After 1989 grazing in spring or winter or increased grazing in summer all increased population sizes. Population sizes fluctuated widely between years. The effects of the grazing treatments and plant sizes on the transitions between nine life-history stages were determined (...)

Moments of the Riemann zeta function

Abstract: Assuming the Riemann hypothesis, we obtain an upper bound for the moments of the Riemann zeta function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in other families of L-functions.

Mass equidistribution for Hecke eigenforms

Abstract: We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors.

Nonvanishing of quadratic Dirichlet L-functions at s=1/2

Abstract: We show that for a positive proportion of fundamental discriminants d, L(1/2,chi_d) != 0. Here chi_d is the primitive quadratic Dirichlet character of conductor d.

The distribution of values of L(1, χ d )

Abstract: Throughout this paper d will denote a fundamental discriminant, and χd the associated primitive real character to the modulus |d|. We investigate here the distribution of values of L(1, χd) as d varies over all fundamental discriminants with |d| ≤ x. Our main concern is to compare the distribution of values of L(1, χd) with the distribution of “random Euler products” L(1,X) = ∏ p(1 − X(p)/p)−1 where the X(p)’s are independent random variables taking values 0 or ±1 with suitable probabilities, described below. For example, we shall give asymptotics for the probability that L(1, χd) exceeds eγτ , and the probability that L(1, χd) ≤ π2 6 1 eγτ uniformly in a wide range of τ . These results are sufficiently uniform to prove slightly more than a recent conjecture of H.L. Montgomery and R.C. Vaughan [MV]. One important motivation for our work is to make progress towards resolving the discrepancy between extreme values that may be exhibited (the omega results of S.D. Chowla described below) and the conditional bounds on these extreme values (the O-results of J.E. Littlewood, see below). The uniformity of our results provides evidence that the omega results of Chowla may represent the true nature of extreme values of L(1, χd). These questions have been studied by many authors, most notably by P.D.T.A. Elliott [E1,2,3] and Montgomery and Vaughan [MV]. We begin by reviewing some of the history of the subject which will help place our results in context. Throughout this paper logj will denote the j-fold iterated logarithm; that is, log2 = log log, log3 = log log log, and so on. In [Li] Littlewood showed that on the assumption of the Generalized Riemann Hypothesis (GRH)

Analytic Number Theory

Abstract: Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

Random Matrix Theory and ζ(1/2+it)

Abstract: We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of |Z| and Z/Z*, and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N→∞. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found the Gaussian limit distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate the leading order N→∞ asymptotics of the moments of |Z| and Z/Z*. These CUE results are then compared with what is known about the Riemann zeta function ζ (s) on its critical line Re s= 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T. Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2+iT) in the limit T→∞. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T. This leads us to a conjecture for the moments of |ζ(1/2+it) |. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles.

Progress in Mathematics

Abstract: Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Herbivory: effects on plant abundance, distribution and population growth.

Abstract: Plants are attacked by many different consumers. A critical question is how often, and under what conditions, common reductions in growth, fecundity or even survival that occur due to herbivory translate to meaningful impacts on abundance, distribution or dynamics of plant populations. Here, we review population-level studies of the effects of consumers on plant dynamics and evaluate: (i) whether particular consumers have predictably more or less influence on plant abundance, (ii) whether particular plant life-history types are predictably more vulnerable to herbivory at the population level, (iii) whether the strength of plant–consumer interactions shifts predictably across environmental gradients and (iv) the role of consumers in influencing plant distributional limits. Existing studies demonstrate numerous examples of consumers limiting local plant abundance and distribution. We found larger effects of consumers on grassland than woodland forbs, stronger effects of herbivory in areas with high versus low disturbance, but no systematic or unambiguous differences in the impact of consumers based on plant life-history or herbivore feeding mode. However, our ability to evaluate these and other patterns is limited by the small (but growing) number of studies in this area. As an impetus for further study, we review strengths and challenges of population-level studies, such as interpreting net impacts of consumers in the presence of density dependence and seed bank dynamics.

Bounded gaps between primes

Abstract: It is proved that lim inf n?8 (p n+1 -p n )<7×10 7 , where p n is the n -th prime. Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose