arXiv subjects
Publications and source records attributed to K. Soundararajan.
We show that there exist arbitrarily large sets $S$ of $s$ prime numbers such that the equation $a+b=c$ has more than $\exp(s^{2-\sqrt{2}-ε})$ solutions in coprime integers $a$, $b$, $c$ all of whose prime factors lie in the set $S$. We also show that there exist sets $S$ for which the equation $a+1=c$ has more than $\exp(s^{\frac 1{16}})$ solutions with all prime factors of $a$ and $c$ lying in $S$.
We give lower bounds of the conjectured order of magnitude for an orthogonal and a symplectic family of L-functions.
For a given $x$ we consider the minimum of $\sum_{n\le x} χ(n)/n$ as $χ$ ranges over all quadratic Dirichlet characters. For all large $x$, this minimum is negative and we give upper and lower bounds for it.
Extending a result of Heath-Brown, we establish an asymptotic formula for the fourth moment of central values of Dirichlet $L$-functions attached to primitive characters $\pmod q$.
In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the GRH. We give a simple proof of their estimate, and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and finds a hidden structure among these characters.
We find the closest approximation to 1 from below using a sum of $n$ Egyptian fractions.
The moments of central values of families of L-functions have recently attracted much attention and, with the work of Keating and Snaith, there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of L-functions. As an example we work out the case of the family of all Dirichlet L-functions to a prime modulus.
Improving a result of N. Levinson, we exhibit large and small values of $|ζ(1+it)|$.
Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of $ψ(x+H)- ψ(x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^δ\le H \le N^{1-δ}$.
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ``well-distributed'' in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equi-distribution, as have Fourier analysts when working with the ``uncertainty principle''. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.
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 authors study the distribution of psi(x+h)-psi(x)-h and compare it with numerical data.
Given a multiplicative function f satisfying |f(n)| <= 1 for all n, the authors study the problem of obtaining explicit upper bounds on the mean-value 1/x |sum_{n <= x} f(n)|.
Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as mean-values of functions in F(s). More precisely, let Gamma_N(S) = {1/N sum_{n <= N} f(n): f in F(S)} and Gamma(S) = lim_{N -> infinity} Gamma_N(s). The authors call Gamma(S) the spectrum of the set S, and study its properties.
The authors study the distribution of zeros of the Fekete polynomial f_p(t) (defined for p prime) as p -> infinity. They show that asymptotically a constant fraction of the zeros lie on the unit circle, and they investigate the constant of proportionality.
We evaluate the character sum sigma_m sigma_n (m/n) where the two sums are of approximately the same length. The answer is surprising.
Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n <= x is o(x); they show that this holds if log x / log log q -> infinity and q -> infinity (q is the size of the finite field).