Related papers: Almost sure large fluctuations of random multiplic…

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the…

Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a_1+...+a_N by s_N, the paper gives the following necessary and sufficient…

The goal of this work is to prove a new sure upper bound in a setting that can be thought of as a simplified function field analogue. This result is comparable to a recent result of the author concerning almost sure upper bound of random…

The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to [2,\infty)$, we determine…

Lacunary function systems of type $(f(M_nx))_{n\geq 1}$ for periodic functions $f$ and sequences of fast-growing matrices $(M_n)_{n\geq 1}$ exhibit many properties of independent random variables like satisfying the Central Limit Theorem or…

In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-\delta}$, for some $\delta>0$, and if the Dirichlet series…

We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg…

Let $f$ be a real-valued $1$-bounded multiplicative function. Suppose that the mean-value of $f^{2}$ exists, and $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n)e^{2\pi i n \alpha} \Big | d \alpha\leq N^{o(1)}$$ as $N \rightarrow \infty$, then…

Let $V(x)$ be the number of sign changes of the partial sums up to $x$, say $M_f(x)$, of a Rademacher random multiplicative function $f$. We prove that the averaged value of $V(x)$ is at least $\gg (\log x)(\log\log x)^{-1/2-\epsilon}$. Our…

Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq…

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the…

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a positive integer and $a$ be an integer with $(a,\,q)=1$. In this paper, we shall prove that $$\sum_{\substack{n\leq N\\ (n,\,q)=1}}f(n)e({a\bar{n}\over…

For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i…

We give an upper bound for the exponential $\sum_{m=1}^M \exp( 2i\pi f (m))$ in terms of $M$ and $\lambda$, where $\lambda$ is a small positive number which denotes the size of the fourth derivative of the real valued function $f$. The…

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…

We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We…

We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$…

It is shown that two conjectures put forward in the recent article Iksanov and Kostohryz (2025) are true. Namely, we prove a functional central limit theorem (FCLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series…

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, $a_k\sim Ck^{p-1}, k\to\infty, p>0$,where $C$ is a positive constant. The measures…