arXiv subjects

Akash Singha Roy

Publications and source records attributed to Akash Singha Roy.

12 recordsLinked to original sources

arXiv2026

The Furstenberg-Sárközy theorem for sums of an even number of odd powers

We obtain a Furstenberg-Sárközy-type result for sets $A\subset [N]$ whose difference set $A-A$ does not contain the sum of $s$-many $k$-th powers of positive integers, with $k>1$ odd and $s>0$ even. Namely, we prove that such sets must satisfy a power-saving bound $|A| \, \ll \, N^{1-\frac1k\min\{s \, σ_k, \, 1/2\}+ε}$ for any fixed $ε>0$, where $σ_k >0 $ is any admissible saving in a classical one-variable Weyl estimate. In particular, we can take $σ_k=\max\left\{2^{1-k}, \, \frac{1}{k(k-1)}\right\}$ using the classical theory and the best currently available bounds for classical Weyl sums. A greedy construction produces a set $A\subset[N]$ with $|A|\gg N^{1-s/k}$ for which $A-A$ contains no sum of $s$-many positive $k$-th powers, so our power-saving bound is of the correct shape.

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arXiv2025

The Landau-Selberg-Delange method for products of Dirichlet $L$-functions, and applications, I

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ of a Dirichlet series $\sum_n \, a_n/n^s$ that behaves like a complex power of the Riemann zeta function. However, situations often arise when the Dirichlet series behaves like a product of complex powers of several Dirichlet $L$-functions to a modulus $q$. In such situations, one often requires sharp asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ that apply in much wider ranges of $q$ than permitted by known forms of the Landau-Selberg-Delange method. In this manuscript, we address this problem, giving new estimates on $\sum_{n \le x} \, a_n$ in ranges of $q$ that are (in most applications) much wider than attainable from previous results. Our results also weaken certain hypotheses on the size of $\{a_n\}_n$. As applications of our main theorems, we extend Landau's classical results on the distribution of integers with prime factors restricted to progressions, and improve upon Chang, Martin and Nguyen's work on the distributions of the least invariant factors and least primary factors of multiplicative groups. We also extend the classical Sathe-Selberg theorem and study the local laws of the functions $Ω_a(n)$ and $ω_a(n)$, that count (with and without multiplicity, respectively), the number of prime divisors of $n$ lying in the progression $a$ mod $q$.

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arXiv2024

Mean values of multiplicative functions and applications to residue-class distribution

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo $q$, where $q$ varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $ϕ(n)$ among the coprime residues to the "correct" moduli $q$ that vary uniformly in a similar range, and also quantify the failure of equidistribution of the values of $ϕ(n)$ among the coprime residue classes to the "incorrect" moduli.

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arXiv2024

Joint distribution in residue classes of families of polynomially-defined multiplicative functions

We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, obtaining analogues of the Siegel--Walfisz Theorem for large classes of multiplicative functions. We extend a criterion of Narkiewicz for families of multiplicative functions that can be controlled by values of polynomials at the first few prime powers, and establish results that are completely uniform in the modulus as well as optimal in most parameters and hypotheses. This also significantly generalizes and improves upon previous work done for a single such function in specialized settings. Our results have applications for most interesting multiplicative functions, such as the Euler totient function $ϕ(n)$, the sum-of-divisors function $σ(n)$, the coefficients of the Eisenstein series, etc., and families of these functions. For instance, an application of our results shows that for any fixed $ε>0$, the functions $ϕ(n)$ and $σ(n)$ are jointly asymptotically equidistributed among the reduced residue classes to moduli $q$ coprime to $6$ varying uniformly up to $(\log x)^{(1-ε)α(q)}$, where $α(q) = \prod_{\ell \mid q} (\ell-3)/(\ell-1)$; furthermore, the coprimality restriction is necessary and the range of $q$ is essentially optimal. One of the primary themes behind our arguments is the quantitative detection of a certain mixing (or ergodicity) phenomenon in multiplicative groups via methods belonging to the `anatomy of integers', but we also rely heavily on more pure analytic arguments (such as a suitable modification of the Landau-Selberg-Delange method), -- whilst using several tools from arithmetic and algebraic geometry, and from linear algebra over rings as well.

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arXiv2023

Joint distribution in residue classes of families of polynomially-defined additive functions

Let $g_1, \dots , g_M$ be additive functions for which there exist nonconstant polynomials $G_1, \dots , G_M$ satisfying $g_i(p) = G_i(p)$ for all primes $p$ and all $i \in \{1, \dots , M\}$. Under fairly general and nearly optimal hypotheses, we show that the functions $g_1, \dots , g_M$ are jointly equidistributed among the residue classes to moduli $q$ varying uniformly up to a fixed but arbitrary power of $\log x$. Thus, we obtain analogues of the Siegel-Walfisz Theorem for primes in arithmetic progressions, but with primes replaced by values of such additive functions. Our results partially extend work of Delange from fixed moduli to varying moduli, and also generalize recent work done for a single additive function.

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arXiv2023

Mean values of multiplicative functions and applications to the distribution of the sum of divisors

We provide uniform bounds on mean values of multiplicative functions under very general hypotheses, detecting certain power savings missed in known results in the literature. As an application, we study the distribution of the sum-of-divisors function $σ(n)$ in coprime residue classes to moduli $q \le (\log x)^K$, obtaining extensions of results of Śliwa that are uniform in a wide range of $q$ and optimal in various parameters. As a consequence of our results, we obtain that the values of $σ(n)$ sampled over $n \le x$ with $σ(n)$ coprime to $q$ are asymptotically equidistributed among the coprime residue classes mod $q$, uniformly for odd $q \le (\log x)^K$. On the other hand, if $q$ is even, then equidistribution is restored provided we restrict to inputs $n$ having sufficiently many prime divisors exceeding $q$.

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arXiv2023

Distribution in coprime residue classes of polynomially-defined multiplicative functions

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue classes of polynomially-defined multiplicative functions, establishing equidistribution results allowing a wide range of uniformity in the modulus $q$. For example, we show that the values $ϕ(n)$, sampled over integers $n \le x$ with $ϕ(n)$ coprime to $q$, are asymptotically equidistributed among the coprime classes modulo $q$, uniformly for moduli $q$ coprime to $6$ that are bounded by a fixed power of $\log{x}$.

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arXiv2023

The distribution of intermediate prime factors

Let $P^{\left(\frac 12\right)}(n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $α\in (0,1)$, the $α$-positioned prime factor of $n$, $P^{(α)}(n)$. It has previously been shown that $\log \log P^{(α)}(n)$ has normal order $α\log \log x$, and its values follow a Gaussian distribution around this value. We extend this work by obtaining an asymptotic formula for the count of $n\leq x$ for which $P^{(α)}(n)=p$, for primes $p$ in a wide range up to $x$. We give several applications of these results, including an exploration of the geometric mean of the middle prime factors, for which we find that $\frac 1x \sum_{1<n \le x} \log P^{\left(\frac 12 \right)}(n) \sim A(\log x)^{φ-1}$, where $φ$ is the golden ratio, and $A$ is an explicit constant. Along the way, we obtain an extension of Lichtman's recent work on the ``dissected'' Mertens' theorem sums $\sum_{\substack{P^+(n) \le y \\ Ω(n)=k}} \frac{1}{n}$ for large values of $k$.

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arXiv2022

On Benford's Law for multiplicative functions

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.

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arXiv2021

Joint distribution in residue classes of polynomial-like multiplicative functions

Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes mod $p$, where $p$ is a growing prime (or nearly prime) modulus. This can be seen as complementary to work of Narkiewicz, who obtained comprehensive results for fixed moduli.

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arXiv2021

Powerfree sums of proper divisors

Let $s(n):= \sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text{$n$ is $k$th powerfree} \Longleftrightarrow \text{$s(n)$ is $k$th powerfree} \] holds almost always (meaning, on a set of asymptotic density $1$). We prove this for $k\ge 4$.

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arXiv2021

Distribution mod $p$ of Euler's totient and the sum of proper divisors

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $ϕ(n)$ and the sum-of-proper-divisors function $s(n):=σ(n)-n$. We prove that the values $ϕ(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.

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