Searcharxiv⌕ Search

arXiv subjects

Kannan Soundararajan

Publications and source records attributed to Kannan Soundararajan.

51 records · Page 3Linked to original sources
arXiv2012

Continuous lower bounds for moments of zeta and L-functions

We obtain lower bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta function for all k > 1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.

math.NT↗
arXiv2011

Riffle shuffles with biased cuts

The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut 'about in half' and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function.

math.PR↗
arXiv2011

Asymptotic Large Sieve

Motivated by applications to the study of L-functions, we develop an asymptotic version of the large sieve inequality for linear forms in primitive Dirichlet characters.

math.NT↗
arXiv2011

Critical zeros of Dirichlet $L$-functions

We use the Asymptotic Large Sieve and Levinson's method to obtain lower bounds for the proportion of simple zeros on the critical line of the twists by primitive Dirichlet characters of a fixed L-function of degree 1,2, or 3.

math.NT↗
arXiv2011

Random multiplicative functions in short intervals

We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.

math.NT↗
arXiv2010

On modular signs

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve significantly the recent estimate of Iwaniec, Kohnen and Sengupta.

math.NT↗
arXiv2009

Bounding |ζ(1/2 + it)| on the Riemann hypothesis

In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |ζ(1/2+it)| \ll \exp(C\log t/\log \log t). In this note we show how the problem of bounding |ζ(1/2+it)| may be framed in terms of minorizing the function \log ((4+x^2)/x^2) by functions whose Fourier transforms are supported in a given interval, and drawing upon recent work of Carneiro and Vaaler we find the optimal such minorant. Thus we establish that any C> (\log 2)/2 is permissible in Littlewood's result.

math.NT↗
arXiv2003

The number of unsieved integers up to x

Typically, one expects that there are around x\prod_{p\not\in P, p <= x} (1-1/p) integers up to x, all of whose prime factors come from the set P. Of course for some choices of P one may get rather more integers, and for some choices of P one may get rather less. Hall [4] showed that one never gets more than e^γ+o(1) times the expected amount (where γis the Euler-Mascheroni constant), which was improved slightly by Hildebrand [5]. Hildebrand [6] also showed that for a given value of \prod_{p\not\in P, p <= x} (1-1/p), the smallest count that you get (asymptotically) is when P consists of all the primes up to a given point. In this paper we shall improve Hildebrand's upper bound, obtaining a result close to optimal, and also give a substantially shorter proof of Hildebrand's lower bound. As part of the proof we give an improved Lipschitz-type bound for such counts.

math.NT↗
arXiv2002

Strong multiplicity one for the Selberg class

In this note we investigate the problem of determining elements of the Selberg class from their Dirichlet series coefficients at the primes. We show that this is possible when the degree is one, but in general need an additional weak hypothesis on the Dirichlet series coefficients at squares of primes.

math.NT↗
arXiv2002

The distribution of values of L(1,chi_d)

In this paper we investigate the distribution of values of L(1,chi) as chi ranges over primitive real characters. In particular we focus on the extent to which this distribution may be approximated by "random Euler products." Our work also establishes a recent conjecture of Montgomery and Vaughan on the frequency of large (or small) values of L(1,chi).

math.NT↗
arXiv2001

Real zeros of quadratic Dirichlet L-functions

We show that for a positive proportion of real primitive Dirichlet characters chi, the associated Dirichlet L-function L(s,chi) has no zeros on the positive real axis. Prior to this it was not known whether or not there were infinitely many L-functions (from any family) with no positive real zeros.

math.NT↗
arXiv2001

Upper bounds for |L(1,chi)|

Given a non-principal Dirichlet character chi mod q, an important problem in number theory is to obtain good estimates for the size of L(1,chi). In this paper we focus on sharpening the upper bounds known for |L(1,chi)|; in particular, we wish to determine constants c (as small as possible) for which the bound |L(1,chi)| <= (c+o(1)) log q holds.

math.NT↗