McCarthy defined a function in terms of quotients of the $p$-adic gamma functions which can best be described as an analogue of hypergeometric series in the $p$-adic setting. We find six transformations and three summation identities for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums. We also find a transformation and two summation identities for the Greene's finite field hypergeometric series. Finally, we obtain some special values of the Greene's finite field hypergeometric series as an application of our main results.

In this paper, we study convergence of nets and sequences of $p$-Bochner, $p$-Dunford and $p$-Pettis integrable functions, $p \in [1, \infty)$, defined on a finite measure space with values in a real Banach space. Applying Hölder's inequality, we study some properties of these functions and convergence of their nets and sequences. We introduce the idea of $\delta$-Cauchy nets in terms of which we establish convergence theorems for nets of above types of functions. We see that $\delta$-Cauchyness of a net plays the similar role as uniform integrability does in case of sequences.

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(\pi(R_n))_{n \geq 1}$, which tells us where the $n$th Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the $n$th Ramanujan prime, which imply an asymptotic formula for $\pi(R_n)$ conjectured by Yang and Togbé. Then we use these explicit estimates to derive a result concerning an inequality involving $\pi(R_n)$ that was conjectured by Sondow, Nicholson, and Noe. In the second part of the paper, we apply the results derived in the first part to obtain some new results concerning the number of Ramanujan primes not exceeding $x$. Finally, we compute the limit of $((R_n - p_{2n})/n)_{n \geq 1}$.

Diophantine problems related to binomial coefficients have a vast literature. In this paper we investigate power and polynomial values of row sums of hyperbolic Pascal triangles, which are recently introduced generalizations of Pascal's classical triangle. We prove various effective and ineffective finiteness results.

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \floor{f(n)}+ y \floor{f(n)}^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $\floor{f(n)}^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\floor{f(n)}^{-1} \pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ \floor{t^c}$ with $c\in(1,\frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.

We study multiplicative functions $f$ satisfying $|f(n)|\le 1$ for all $n$, the associated Dirichlet series $F(s):=\sum_{n=1}^{\infty} f(n) n^{-s}$, and the summatory function $S_f(x):=\sum_{n\le x} f(n)$. Up to a possible trivial contribution from the numbers $f(2^k)$, $F(s)$ may have at most one zero or one pole on the one-line, in a sense made precise by Hal\'{a}sz. We estimate $\log F(s)$ away from any such point and show that if $F(s)$ has a zero on the one-line in the sense of Halász, then $|S_f(x)|\le (x/\log x) \exp\big(c\sqrt{\log \log x}\big)$ for all $c>0$ when $x$ is large enough. This bound is best possible.

In this note we show a new bound for the divisor function. This improves a result, given by Mardjanichvili.