Related papers: Legendre's formula and $p$-adic analysis

Let $p$ be a prime, $\varepsilon>0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\varepsilon}< N <p^{1-\varepsilon}$, then $$ \#\{n!\!\!\! \pmod p;\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon)>0. $$ We use this bound to…

Let $p>5$ be a prime. Motivated by the known formulae $\sum_{k=1}^\infty(-1)^k/(k^3\binom{2k}{k})=-2\zeta(3)/5$ and $\sum_{k=0}^\infty \binom{2k}{k}^2/((2k+1)16^k)=4G/\pi$$ (where $G=\sum_{k=0}^\infty(-1)^k/(2k+1)^2$ is the Catalan…

In a previous paper the second author developed a new approach to the abelian p-adic Stark Conjecture at s=1 and stated some related conjectures. This paper develops and applies techniques using p-adic measures and continued fractions to…

We construct $p$-adic $L$-functions interpolating critical $L$-values of algebraic Hecke characters for arbitrary unramified primes $p$ and any totally imaginary field. For non-ordinary primes, the only previously known case was that of…

Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and…

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace $p$ by a set of primes pi and prove a pi-version of…

$p$-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this survey, we will review $p$-adic Hodge Theory for algebraic varieties, present current developments in $p$-adic Hodge Theory for analytic…

A linear code is said to be $\Delta$-divisible if the Hamming weights of all its codewords are divisible by $\Delta$. The $p$-adic valuation of a code is defined as the greatest integer $t$ such that the code is $p^t$-divisible. In this…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times…

If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{\sigma(p^k)}{p^k}$ and $\frac{\sigma(m^2)}{m^2}$, and the ratios…

Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\frac {j+k}p)\pm(\frac{j-k}p)$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. When $p\equiv1\pmod4$, we determine the characteristic polynomials…

We present a concise method for deriving an explicit formula for $p$-adic multiple zeta values. The formula features a variant of multiple harmonic sums, termed binomial multiple harmonic sums.

In this paper we have discussed convergence of power series both in p-adic norm as well as real norm. We have investigated rational summability of power series with respect to both p-adic norm and real norm under certain conditions. Then we…

For an odd prime $p$ and a positive integer $n$, let ${_n}G_n[\cdots]_p$ denote McCarthy's $p$-adic hypergeometric function. In this article, we prove $p$-adic analogue of certain classical hypergeometric identities and using these…

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related…

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have…

We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial…

We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective…

In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det\left[x+\left(\frac{i-j}p\right)+\left(\frac ip\right)y+\left(\frac jp\right)z+\left(\frac{ij}p\right)w\right]_{0\le i,j\le(p-3)/2}$$ for any odd…