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Nilotpal Sinha

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Preprint Aug 2026

Chebyshev Bias for Largest Prime Factors

Let $P^+(n)$ be the largest prime factor of $n$, and let $\chi=\chi_{-4}$ be $1$ on primes $1\pmod4$ and $-1$ on primes $3\pmod4$. For fixed $k\ge2$ we study \[ D_k(x)=\sum_{\substack{n\le x\\ \Omega(n)=k}}\chi(P^+(n)). \] Thus $D_k(x)$ compares the two residue classes according to the largest prime factor of integers having exactly $k$ prime factors, counted with multiplicity. Assuming RH for $\zeta(s)$ and $L(s,\chi)=\beta(s)$, we prove a pointwise explicit formula. The main term is a fixed negative contribution plus an absolutely convergent oscillating sum over the zeros $\rho=\frac12+i\gamma$ of $L(s,\chi)$. The ordering of the prime factors produces $k$ Perron denominators, and the principal coefficient of a zero is $O_k((1+|\gamma|)^{-k})$. We then show that the total size of all zero terms is strictly smaller than the fixed contribution. Hence $D_k(x)<0$ for all sufficiently large $x$. The analogous problem without fixing $k$ is still open.

Nilotpal Sinha · 0 citations

Preprint Aug 2026

Asymptotic Formulae For Reciprocal Partitions

For integers $1\le m\le n$, let $s(n,m)$ denote the number of $m$-tuples $(k_1,\ldots,k_m)$ of nonnegative integers satisfying \[ n=\sum_{j=1}^{m}\frac{k_j}{j}. \] We obtain a complete asymptotic expansion for $\log s(n,m)$, uniformly for all $n\ge m$ as $m\to\infty$. The coefficients are given explicitly in terms of limiting prime-block functions arising from the residue structure of the problem. We also determine the full hierarchy of multiplicative corrections in the sparse regime, identifying explicit constants at every fixed order and, in particular, the first correction constants $1/4$ and $(1-\log 2)/4$. The results give uniform two-parameter asymptotics as both the target and the number of allowed reciprocal parts grow.

Nilotpal Sinha · 0 citations