Related papers: The distribution of $d_4(n)$ in arithmetic progres…

We say that $d_3(n)$ has exponent of distribution $\theta$ if, for every $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{\theta-\varepsilon}$. Nguyen proved, following earlier work of Banks, Heath-Brown,…

Nguyen has shown that on averaging over $a=1,...,q$ the 3-fold divisor function has exponent of distribution 2/3, following \cite {banks}. We follow [2] which leads to stronger bounds.

We study the average value of the divisor function $\tau(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$.…

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. For…

We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In…

We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-\epsilon}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of…

We prove a Lindel\"of-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the least positive integer such that…

We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6,…

We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture…

We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$ R(n) \leq O(\psi(n)n^{\frac{1}{4}}) $$ where $$ R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor -…

For Dirichlet series roughly of the type $Z(s, w) = sum_d L(s, chi_d) d^{-w}$ the subconvexity bound $Z(s, w) \ll (sw(s+w))^{1/6+\varepsilon}$ is proved on the critical lines $\Re s = \Re w = 1/2$. The convexity bound would replace 1/6 with…

We introduce a novel arithmetic function $w(n)$, a generalization of the Liouville function $\lambda(n)$, as the coefficients of a Dirichlet series. By spatially encoding information in a natural way about the distribution of prime factors…

A constrained diffusive random walk of n steps and a random flight in Rd, which can be expressed in the same terms, were investigated independently in recent papers. The n steps of the walk are identically and independently distributed…

An approach will be proposed to determine the existence of a limit distribution of additive arithmetic functions in this work. It is based on assertions that will be proven in this work and on the properties of Dirichlet convolution and…

A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_{\epsilon}n^{1+\epsilon}$ times (for any $\epsilon>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the…

We develop a general method for lower bounding the variance of sequences in arithmetic progressions mod $q$, summed over all $q \leq Q$, building on previous work of Liu, Perelli, Hooley, and others. The proofs lower bound the variance by…

The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these…

We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.

This is a companion paper to arXiv:2312.10772. We deduce an equidistribution theorem for periodic nilsequences and use this theorem to give two applications in arithmetic combinatorics. The first application is quasi-polynomial bounds for a…

For $q \geq 2$, $n \in \mathbb{N}$, let $s_{q}(n)$ denote the sum of the digits of $n$ written in base $q$. Spiegelhofer (2020) proved that the Thue--Morse sequence has level of distribution $1$, improving on a former result of Fouvry and…