arXiv subjects

Kunjakanan Nath

Publications and source records attributed to Kunjakanan Nath.

6 recordsLinked to original sources

arXiv2026

The distribution of the maximum of cubic character sums

For a primitive Dirichlet character $χ\pmod q$ we let \[M(χ):= \frac{1}{\sqrt{q}}\max_{1\leq t \leq q} \Big|\sum_{n \leq t} χ(n) \Big|.\] In this paper, we investigate the distribution of $M(χ)$, as $χ$ ranges over primitive cubic characters $χ\pmod q$ with $(q,3)=1$ and $q\leq Q$. Our first result gives an estimate for the proportion of such characters for which $M(χ)>V$, in a uniform range of $V$, which is best possible under the assumption of the Generalized Riemann Hypothesis. In particular, we show that the distribution of large cubic character sums behaves very differently from those in the family of non-principal characters modulo a large prime, and the family of quadratic characters. We also investigate the location of the number $N_χ$ where the maximum of $|\sum_{n\leq N} χ(n)|$ is attained, and show the surprising result that for almost all primitive cubic characters $χ\pmod q$ with $M(χ)>V$, $N_χ/q$ is very close to a reduced fraction with a large denominator of size $(\log V)^{1/2+o(1)}$. This contradicts the common belief that for an even character $χ$, $N_χ/q$ is located near a rational of small denominator and gives a striking difference with the case of even characters in the other two families mentioned above, for which $N_χ/q\approx 1/3$ or $2/3$ for almost all even $χ$. Furthermore, in the case of cubic characters, the works of Granville-Soundararajan, Goldmakher, and Lamzouri-Mangerel show that if $M(χ)$ is large, then $χ$ pretends to be $ξ(n)n^{it}$ for some small $t$, where $ξ$ is an odd character of small conductor $m$. We show that for almost all such characters, we have $M(χ)=m^{-1/2+o(1)}\big|L(1+it, χ\overlineξ)\big|.$

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arXiv2026

Almost primes and primes that are sums of two squares plus 1

In this paper, we obtain a lower bound for the number of primes $p\leq x$ such that $p-1$ is a sum of two squares and $p+2$ has a bounded number of prime factors. The proof uses the vector sieve framework, involving a semi-linear sieve and a linear sieve.

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arXiv2026

Diophantine approximation with integers having no large prime factors

Given any irrational number $α$, we show that for any $0<θ<6/17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\|nα\| < n^{-θ},$$ where $(\log n)^C\leq y\leq n$ for some large constant $C>0$. This improves the previous work of Baker, who obtained the exponent $1/3-2/(3C)+o(1)$ in the case of $y\geq (\log n)^C$, and that of Yau, who obtained the exponent $1/3$ when $y=n^{o(1)}$. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers.

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arXiv2026

Real zeros of $L'(s, χ_d)$

In 1990, Baker and Montgomery conjectured that $L'(s,χ_d)$ has $\asymp \log\log |d|$ real zeros in the interval $[1/2,1]$ for almost all fundamental discriminants $d$. The study of these zeros was motivated by their connection to real zeros of Fekete polynomials and to sign changes of the character sums $\sum_{n\leq x}χ_d(n)$. Recent work of Klurman, Lamzouri, and Munsch shows that the number of such zeros is $\gg (\log\log |d|)/(\log\log\log\log |d|)$ for almost all $d$, thereby establishing the conjectured lower bound up to the factor $\log\log\log\log |d|$. In this paper, we prove that for almost all fundamental discriminants $d$, $L'(s,χ_d)$ has at most $(\log\log |d|)(\log\log\log |d|)$ real zeros in $[1/2,1]$, thus resolving the Baker-Montgomery conjecture up to a factor of $\log\log\log |d|$. We also give a quantitative upper bound on the exceptional set of discriminants. Furthermore, we show, conditionally on certain natural assumptions, that $100\%$ of these zeros lie away from $1/2$.

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arXiv2025

On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments

Let $\{λ_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, χ)$, where $χ$ is a Dirichlet character modulo $q$, that if $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ with $\varepsilon>0$, then for any $A\geq 1$, $$\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A} \frac{HX^2}{(\log X)^{A}}$$ holds. Moreover, under an additional hypothesis on the fourth moment of certain Dirichlet polynomials (which follows from GRH for $L(s, f)$), we show that the above result can be strengthened to hold in a wider range $X^{1/3+\varepsilon}\ll H \ll X^{1-\varepsilon}$. Finally, if we average over the forms $f$, then for $X^{\varepsilon}\ll H\ll X^{1-\varepsilon}$ and for any $A\geq 1$, $$ \sum_{f\in \mathcal{H}_k}ω_f\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A}\frac{HX^2}{(\log X)^{A}},$$ where $\mathcal{H}_k$ is the Hecke basis for the space of holomorphic cusp forms of weight $k$ for the full modular group $\mathrm{SL}(2, \mathbb{Z})$ and $ω_f$ are harmonic weights associated with $f\in \mathcal{H}_k$. These results may be viewed as modular analogues of the averaged forms of the Hardy--Littlewood prime tuple conjecture.

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arXiv2023

Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their $b$-adic expansion for some large positive integer $b$. The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi-linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form $p=1+m^2+n^2$ with a missing digit in a large odd base $b$.

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