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Register a new userAsymptotics of the number of partitions into p-cores and some trigonometric sums
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Authors
Gert Almkvist
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An asymptotic formula for the number of partitions into p-cores is derived. As a byproduct some integer valued trigonometric sums are found
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In recent work, M. Schneider and the first author studied a curious class of integer partitions called sequentially congruent partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the number of parts. Let $p_{mathcal S}(n)$ be the number of sequentially congruent partitions of $n,$ and let $p_{square}(n)$ be the number of partitions of $n$ wherein all parts are squares. In this note we prove bijectively, for all $ngeq 1,$ that $p_{mathcal S}(n) = p_{square}(n).$ Our proof naturally extends to show other exotic classes of partitions of $n$ are in bijection with certain partitions of $n$ into $k$th powers.
The paper compares the asymptotic of the expressions $frac {1} {x} sumlimits_{n leq x} {f(n)}$ and $sumlimits_{n leq x} {frac {f(n)} {n}}$, $frac {1} {x} sumlimits_{p leq x} {f(p)}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$. The asymptotic of sums $sumlimits_{n leq x} {frac {f(n)} {n}}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$ ($n,p$ - respectively, positive and prime numbers) are determined if the asymptotic of sums are known, respectively: $sumlimits_{n leq x} {f(n)}$,$sumlimits_{p leq x} {f(p)}$.
Let $kappa$ be a positive real number and $minmathbb{N}cup{infty}$ be given. Let $p_{kappa, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(lfloor ell^{kappa}rfloor)_{ellin mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan type for $p_{kappa, m}(n)$, by employing a framework of asymptotics of partitions established by Roth-Szekeres in 1953, as well as some results on equidistribution.
We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=sum_{x in F} psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $psi$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. We exclude exponents $d$ that are congruent to a power of $p$ modulo $|F^times|$ (degenerate $d$), which yield trivial Weil sums. We prove that $V_{F,d} leq (2/3)[Fcolon{mathbb F}_p]$ for any $F$ and any nondegenerate $d$, and prove that this bound is actually reached in infinitely many fields $F$. We also prove some stronger bounds that apply when $[Fcolon{mathbb F}_p]$ is a power of $2$ or when $d$ is not congruent to $1$ modulo $p-1$, and show that each of these bounds is reached for infinitely many $F$.
In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $R_n(p,t)=sum olimits_{m=0}^n m^p H_m^t$, where $H_m$ are harmonic numbers. When $tle 4$ these sums were first studied by Spiess around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums $R_n(p,t)$ for all $tge 0$.
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