arXiv subjects

Priyamvad Srivastav

Publications and source records attributed to Priyamvad Srivastav.

6 recordsLinked to original sources

arXiv2026

Log-free bounds on exponential sums over primes

We establish completely log-free bounds for exponential sums over the primes and the Möbius function. Let $0<η\leq 1/10$, and suppose $α= a/q + δ/x$, with $(a,q)=1$ and $|δ| \leq x^{1/5 + η}/q$, and set $δ_0 = \max(1, |δ|/4)$. For $x \geq x_0(η)$ sufficiently large, we show that: \begin{equation*} \Biggl| \sum_{n \leq x} Λ(n) e(nα) \Biggr| \leq \frac{q}{φ(q)} \frac{\mathscr{F}_η\bigl( \frac{\log δ_0 q}{\log x}, \frac{\log^+ δ_0/q}{\log x} \bigr) \cdot x }{\sqrt{δ_0 q}} \ \text{ and } \ \Biggl| \sum_{n \leq x} μ(n) e(nα) \Biggr| \leq \frac{\mathscr{G}_η\bigl( \frac{\log δ_0 q}{\log x}, \frac{\log^+ δ_0/q}{\log x} \bigr) \cdot x}{\sqrt{δ_0 φ(q)}}, \end{equation*} for all $1 \leq q \leq x^{2/5 - η}$, where $\log^+ z = \max(\log z, 0)$, and the functions $\mathscr{F}_η$ and $\mathscr{G}_η$ are explicitly determined, taking small to moderate values. These bounds improve substantially upon the existing results - particularly with respect to the permissible ranges of $q$, $δ$ in which log-free bounds are known to hold and potentially with respect to asymptotic functions $\mathscr{F}_η$ and $\mathscr{G}_η$ as well. Moreover, the range $1 \leq q \leq x^{2/5 - η}$ is essentially the best possible we can expect. The main innovation is a sieve-weighted version of Vaughan's identity (Lemma 2.1), which is effectively log-free. We employ several ideas and results from the pioneering work of Helfgott, and particularly, they play a central role in ensuring the log-freeness of the type-I contribution. Also, like in his work, these bounds improve as $δ$ increases.

math.NT↗

arXiv2024

Convolution of periodic multiplicative functions and the divisor problem

We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and $1$-pretentious. We prove that if $f_1$ and $f_2$ belong to this class, then $\sum_{n\leq x}(f_1\ast f_2)(n)=Ω(x^{1/4})$. This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between $Δ(x)$ and $Δ(θx)$, where $θ$ is a fixed real number. We prove that there is a non-trivial correlation when $θ$ is rational, and a decorrelation when $θ$ is irrational. Moreover, if $θ$ has a finite irrationality measure, then we can make it quantitative this decorrelation in terms of this measure.

math.NT↗

arXiv2022

Product of three primes in large arithmetic progressions

For any $ε>0$, there exists $q_0(ε)$ such for any $q\ge q_0(ε)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+ε}$. If we restrict our attention to odd moduli $q$ that do not have prime factors congruent to 1 mod 4, we can find such primes below $q^{\frac{11}{8}+ε}$. If we further restrict our set of moduli to prime $q$ that are such that $(q-1,4\cdot7\cdot11\cdot17\cdot23\cdot29)=2$, we can find such primes below $q^{\frac{6}{5}+ε}$. Finally, for any $ε>0$, there exists $q_0(ε)$ such that when $q\ge q_0(ε)$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly four primes, all of which are below $q(\log q)^6$.

math.NT↗

arXiv2016

On correlations of certain multiplicative functions

In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$ for any $|h| \leq x/2$ and arithmetic functions $F=f*1$ and $G=g*1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of magnitude of $f$ and $g$. We also provide asymptotic formula for sums of the type $\sum\limits_{n \leq x} μ^2(n) G(n-h)$, where $G=g*1$ and $g$ is small. For small order of magnitudes of $f$ and $g$, we improve the error terms and make them independent of $h$.

math.NT↗

arXiv2016

On the number of factorizations of an integer

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log x}{\log \log x}} \left( 1 + o(1) \right) \right)$, where $C=2π\sqrt{2/3}$ and $x$ is sufficiently large. This improves upon a previous result of the first author and F. Luca.

math.NT↗

arXiv2015

On Selberg's approximation to the twin prime problem

In his Classical approximation to the Twin prime problem, Selberg proved that for $x$ sufficiently large, there is an $n \in (x,2x)$ such that $2^{Ω(n)}+2^{Ω(n+2)} \leq λ$ with $λ=14$, where $Ω(n)$ is the number of prime factors of $n$ counted with multiplicity. This enabled him to show that for infinitely many $n$, $n(n+2)$ has atmost $5$ prime factors, with one having atmost $2$ and the other having atmost $3$ prime factors. By adopting Selberg's approach and using a refinement suggested by Selberg, we improve this value of $λ$ to about $λ=12.59$.

math.NT↗