arXiv subjects
Publications and source records attributed to K. Soundararajan.
This paper describes a method to compute lower bounds for moments of $ζ$ and $L$-functions. The method is illustrated in the case of moments of $|ζ(\frac 12+it)|$, where the results are new for small moments $0< k<1$.
This is an expository note discussing how the Erdos--Ramanujan proof of Bertrand's postulate may be adapted to show the existence of finite fields.
The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on average only if $v=0,-1,-2,\dots$. Moreover, if $v<0$, then the Dirichlet series associated to $f$ must have a zero of multiplicity $-v$ at $s=1$. In this paper, we prove a converse result that shows that if $f$ is a multiplicative function that is bounded by a suitable divisor function, and $f$ has very small partial sums, then there must be finitely many real numbers $γ_1$, $\dots$, $γ_m$ such that $f(p)\approx -p^{iγ_1}-\cdots-p^{-iγ_m}$ on average. The numbers $γ_j$ correspond to ordinates of zeroes of the Dirichlet series associated to $f$, counted with multiplicity. This generalizes a result of the first author, who handled the case when $|f|\le 1$ in previous work.
This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigenforms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants $D$ such that the Fourier coefficients evaluated at $|D|$ are non-zero. By adapting the resonance method, we also demonstrate that such Fourier coefficients must take quite large values.
Motivated by a question of Venkataramana, we consider the greatest common divisor of $ϕ(f(n))$ where $f$ is a primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, we establish that this gcd may be bounded just in terms of the degree of the polynomial $f$. Unconditionally we establish such a bound for quadratic polynomials, as well as polynomials that split completely into linear factors. The paper also addresses a question of Calegari, and establishes that there are infinitely many $n$ such that $n^2+1$ is not divisible by any prime $\equiv 1 \bmod 2^m$ provided $m$ is a large fixed integer.
This paper discusses examples of integral factorial ratios of height 2 or more. It classifies (apart from finitely many examples) such factorial ratios with height 2 and norm at most 1/3, and describes a general result which exhibits more than 50 two parameter families of integral factorial ratios with height 2. These examples are shown to be irreducible in the sense that they do not arise from multiplying two factorial ratios of height 1.
This paper describes a new approach to classifying integral factorial ratio, obtaining in particular a direct proof of a result of Bober. These results generalize an observation going back to Chebyshev that $(30n)!n!/((15n)!(10n)!(6n)!)$ is an integer for all $n$. Due to the work of Rodriguez-Villegas and Beukers and Heckman, this problem is closely related to classifying hypergeometric functions with finite monodromy groups, and the result of Bober was originally derived as a consequence of the work of Beukers--Heckman. The new proof is elementary and makes partial progress on other related questions.
A classical result of Erdos and Turan states that if a monic polynomial has small size on the unit circle and its constant coefficient is not too small, then its zeros cluster near the unit circle and become equidistributed in angle. Using Fourier analysis we give a short and self-contained proof of this result.
We prove a sharp version of Halász's theorem on sums $\sum_{n \leq x} f(n)$ of multiplicative functions $f$ with $|f(n)|\le 1$. Our proof avoids the "average of averages" and "integration over $α$" manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method we express $\sum_{n \leq x} f(n)$ as a triple Dirichlet convolution, and apply Perron's formula.
Halász's Theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\le 1$ has either mean value $0$, or is "close to" $n^{it}$ for some fixed $t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to short intervals and to arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel's Theorem), and that there are always primes near to the start of an arithmetic progression (Linnik's Theorem).
We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Polyá-Vinogradov gap" in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform, and we give applications related to trace functions over finite fields.
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.
We prove a formula, with power savings, for the sixth moment of Dirichlet L-functions averaged over moduli $q$, over primitive characters $χ$ modulo $q$, and over the critical line. Our formula agrees precisely with predictions motivated by random matrix theory. In particular, the constant 42 appears as a factor in the leading order term, exactly as is predicted for the sixth moment of the Riemann zeta-function.
This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Riemann Hypothesis (GRH) implies there are infinitely many primitive solutions when p>8. We sketch some details of the proof of the latter result.
This paper studies integer solutions to the Diophantine equation A+B=C in which none of A, B, C have a large prime factor. We set H(A, B,C) = max(|A|, |B|, |C|), and consider primitive solutions (gcd}(A, B, C)=1) having no prime factor p larger than (log H(A, B,C))^K, for a given finite K. On the assumption that the Generalized Riemann hypothesis (GRH) holds, we show that for any K > 8 there are infinitely many such primitive solutions having no prime factor larger than (log H(A, B, C))^K. We obtain in this range an asymptotic formula for the number of such suitably weighted primitive solutions.
We study the distribution of closed geodesics for the modular surface. We improve the error term in the prime geodesic theorem, and obtain results on prime geodesics in very short intervals conditionally on the generalized Riemann Hypothesis for Dirichlet L-functions. We emphasize a connection between the closed geodesics and certain Dirichlet L-functions.
Assuming the Riemann hypothesis we demonstrate the existence of smooth numbers in certain short intervals.
We prove a conjecture of Broadurst (arXiv:1004.0519v1) on asymptotic expansions of certain polylogarithm type functions related to the Dickman function.