Related papers: Odd values of the Klein j-function and the cubic p…

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…

Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (reps. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound, we show that…

For completely multiplicative functions f(n) taking values 1 and -1, under certain conditions on f(n) we show that f(n) changes sign at least x exp(-7(log log x)sqrt(log x)) times as n runs through the integers <= x.

For an odd integer $d > 1$ and a finite Galois extension $K/\mathbb{Q}$ of degree $d$, G. L\"{u} and Z. Yang \cite{lu3} obtained an asymptotic formula for the mean values of the divisor function for $K$ over square integers. In this…

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on $\ell$-adic properties…

We study the asymptotics of Schur polynomials with partitions $\lambda$ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots,(m-1),0)$ by at most one component at the beginning as…

In recent work, Amdeberhan and Merca considered the integer partition function $a(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts…

We study the complexity of optimizing highly smooth convex functions. For a positive integer $p$, we want to find an $\epsilon$-approximate minimum of a convex function $f$, given oracle access to the function and its first $p$ derivatives,…

In this paper, we establish that the number of partitions of a natural number with positive odd rank is equal to the number of two-color partitions (red and blue), where the smallest part is even (say $2n$) and all red parts are even and…

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function $x$, and such that whenever $f$ and $g$ are in the set, $f+g,fg$ and…

Let $\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\ge…

For fixed $m$ and $R\subseteq \{0,1,\ldots,m-1\}$, take $A$ to be the set of positive integers congruent modulo $m$ to one of the elements of $R$, and let $p_A(n)$ be the number of ways to write $n$ as a sum of elements of $A$. Nathanson…

In this paper, we give explicit error bounds for the asymptotic expansion of the shifted distinct partition function $q(n +s)$ for any nonnegative integer $s$. Then based on this refined asymptotic formula, we give the exact thresholds of…

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant…

We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials $\Phi_N$ for the elliptic $j$-function. These bounds make explicit the best previously known asymptotic bounds. We then give an explicit…

We show that the number of non-isotopic commutative semifields of odd order $p^{n}$ is exponential in $n$ when $n = 4t$ and $t$ is not a power of $2$. We introduce a new family of commutative semifields and a method for proving isotopy…

An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv…

Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(\lambda)$ for an integer partition $\lambda$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $\lambda$. They then…

In 2010 Zagier introduced the notion of a quantum modular form. One of his first examples was the "strange" function $F(q)$ of Kontsevich. Here we produce a new example of a quantum modular form by making use of some of Ramanujan's mock…

Building on work of Hardy and Ramanujan, Rademacher proved a well-known formula for the values of the ordinary partition function $p(n)$. More recently, Bruinier and Ono obtained an algebraic formula for these values. Here we study the…