Related papers: Sums of class numbers and mixed mock modular forms

Let $p$ be an odd prime and let $d\in\{2,3,7\}$. When $(\frac{-d}p)=1$ we can write $p=x^2+dy^2$ with $x,y\in\mathbb Z$; in this paper we aim at determining $x$ or $y$ modulo $p^2$. For example, when $p=x^2+3y^2$, we show that if $p\equiv…

For any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and…

We define A_n=\sum_{i=1}^n (-1)^i\frac{1}{i} and we show that, for every prime p, there exists a number n such that A_n\equiv 0 (mod p).

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example,…

We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value…

The Ap\'ery polynomials are given by $$A_n(x)=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2x^k\ \ (n=0,1,2,\ldots).$$ (Those $A_n=A_n(1)$ are Ap\'ery numbers.) Let $p$ be an odd prime. We show that…

For any number $m \equiv 0,1 \, (4)$ we correct the generating function of Hurwitz class number sums $\sum_r H(4n - mr^2)$ to a modular form (or quasimodular form if $m$ is a square) of weight two for the Weil representation attached to a…

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$…

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been…

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/n$ be the $n$th harmonic number. In this note we prove that for any prime $p\ge 7$, $$ \sum_{k=1}^{p-1}\frac{H_k}{k^2}\equiv \sum_{k=1}^{p-1}\frac{H_k^2}{k}…

We prove a formula for weighted sums of the first $n$ coefficients of mock modular forms of moderate growth and apply it to Hurwitz class numbers and coefficients of negative half integral weight Eisenstein series, which take the form of…

Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $\nu_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ and $\nu_p(\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k})$ are at…

We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi…

We express a family of Hecke--Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. Hikami and Lovejoy's family of…

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p.…

If $p$ is an odd prime, then we prove that $\e(H_2(G,\mathbb{Z})) \mid p\ \e(G)$ for $p$ groups of class 7. We prove the same for $p$ groups of class at most $p+1$ with $\e(Z(G))=p$. We also prove Schurs conjecture if $\e(G/Z(G))$ is $2,3$…

We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…

In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv…