arXiv subjects

Cihan Sabuncu

Publications and source records attributed to Cihan Sabuncu.

8 recordsLinked to original sources

arXiv2026

The multiplication table problem in large dimensions

For $N\geq 2$ and $k\geq 1$, let $M_k(N):=\#\{x_1\cdots x_k : x_i\in\{1,\ldots,N\}\text{ for all } i\}$ be the $k$-dimensional multiplication table. Given $N$, Khovanskii's theorem implies that $M_k(N)$ agrees, for all sufficiently large $k$, with a polynomial in $k$ of degree $\pi(N)$. We determine the asymptotic size of its leading coefficient, proving that, as $N\to\infty$, with $k$ sufficiently large relative to $N$, \[ M_k(N) = \exp\bigg((2\pi+o(1))\frac{\sqrt{N}}{\log N}\bigg)\frac{k^{\pi(N)}}{\pi(N)!}. \] We also study the analogous problem when the factors are restricted to $y$-smooth integers. For $y=o(\log N)$, we prove that the number of distinct products of $k$ such integers up to $N$ is asymptotic to the number of $y$-smooth integers up to $N^k$, uniformly for $k\geq 1$.

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arXiv2026

Ranks of Elliptic Curves Twisted by Quadratic Forms

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $E^d$ be its twist by the quadratic character $\chi_d$. We prove there are infinitely many twists $d$ which are sums of two squares such that $E^d$ has rank $1$. This result is achieved using moments of derivatives of modular $L$-functions, and particularly captures the lower derivatives which were left out in the work of Munshi. Such a result, in particular, also gives us information on the elliptic fibration $(1+t^2)y^2=f(x)$, where $f(x)$ is a cubic polynomial.

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arXiv2026

Missing digits and sums of two prime squares

We investigate integers whose base $g$ expansion omits a fixed digit and which can be represented as a sum of two prime squares. In the first part of the paper, we apply the Hardy--Littlewood circle method to obtain asymptotic formulas for weighted count of representations of such integers up to $g^k$ as $k\to\infty$, where we weight by the von Mangoldt function. In this case, we also get an interesting bias depending on the fixed digit we are missing. In the second part, combining the circle method with sieve methods, we study the second moment of the corresponding unweighted counting function. This allows us to get a nontrivial lower bound for the cardinality of the set $$\{ n \leq g^k : n \text{ omits the digit } b \text{ in its base } g \text{ expansion and } n = p^2 + q^2 \text{ for some primes } p,q \}. $$

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arXiv2024

On extreme values of $r_3(n)$ in arithmetic progressions

For a given integer $m$ and any residue $a \pmod{m}$ that can be written as a sum of 3 squares modulo $m$, we show the existence of infinitely many integers $n \equiv a \pmod{m}$ such that the number of representations of $n$ as a sum of three squares, $r_3(n)$, satisfies $r_3(n) \gg_m \sqrt{n} \log \log n$. Consequently, we establish that there are infinitely many integers $n \equiv a \pmod{m}$ for which the Hurwitz class number $H(n)$ also satisfies $H(n) \gg_m \sqrt{n} \log \log n$.

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arXiv2024

Visiting early at prime times

Given an integer $m \geq 2$ and a sufficiently large $q$, we apply a variant of the Maynard--Tao sieve weight to establish the existence of an arithmetic progression with common difference $q$ for which the $m$-th least prime in such progression is $\ll_m q$, which is best possible. As we vary over progressions instead of fixing a particular one, the nature of our result differs from others in the literature. Furthermore, we generalize our result to dynamical systems. The quality of the result depends crucially on the first return time, which we illustrate in the case of Diophantine approximation.

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arXiv2024

Right-angled triangles with almost prime hypotenuse

The sequence OEIS A281505 consists of distinct odd legs in right triangles with integer sides and prime hypotenuse. In this paper, we count the closely related quantity of even legs with almost prime hypotenuse. More precisely, we obtain the correct order of magnitude upper and lower bounds for the set of distinct even legs with 5-almost prime hypotenuse. This is a strong version of the appropriate analogy to a conjecture of Chow and Pomerance (stated there for prime hypotenuse).

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arXiv2023

The multiplication table constant and sums of two squares

We will show that the number of integers $\leq x$ that can be written as the square of an integer plus the square of a prime equals $\frac{\pi}{2} \cdot \frac {x}{\log x}$ minus a secondary term of size $x/(\log x)^{ 1+\delta+o(1)}$, where $\delta := 1 - \frac{1+\log\log 2}{\log 2} = 0.0860713320\dots$ is the multiplication table constant. Detailed heuristics suggest that this secondary term is asymptotic to $\frac{1 }{\sqrt{\log\log x}} \cdot \frac x{(\log x)^{ 1+\delta}} $ times a bounded, positive, $1$-periodic, non-constant function of $\frac{\log\log x}{\log 2}$.

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arXiv2023

On the Moments of the Number of Representations as Sums of Two Prime Squares

We study the moments of the function that counts the number of representations of an integer as sums of two prime squares. We refine some of the previous arguments and apply the Selberg sieve to get an unconditional upper bound for all moments. We also prove a lower bound for all moments conditional on some generalization of the Green-Tao theorem on linear equations in primes. More precisely, for the fifth moment and onward, we get the expected order of magnitude lower and upper bounds. In addition, we provide some heuristics on the mass function of this representation function.

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