In the preceding papers, two of authors developed criteria for Greenberg conjecture of the cyclotomic $\mathbb{Z}_2$-extension of $k=\mathbb{Q}(\sqrt{p})$ with prime number $p$. Criteria and numerical algorithm in [5], [3] and [6] enable us to show $\lambda_2(k)=0$ for all $p$ less than $10^5$ except $p=13841, 67073$. All the known criteria at present can not handle $p=13841, 67073$. In this paper, we develop another criterion for $\lambda_2(k)=0$ using cyclotomic units and Iwasawa polynomials, which is considered a slight modification of the method of Ichimura and Sumida. Our new criterion fits the numerical examination and quickly shows that $\lambda_2(\mathbb{Q}(\sqrt{p}))=0$ for $p=13841, 67073$. So we announce here that $\lambda_2(\mathbb{Q}(\sqrt{p}))=0$ for all prime numbers $p$ less that $10^5$.

We give an algebraic description of involutions in the Riordan group.

In the paper, we develop an approach to evaluation of Euler sums that involve harmonic numbers and alternating harmonic numbers. We give explicit formulae for several classes of Euler sums in terms of Riemann zeta values and prove that the quadratic sums ${S_{{l^2},l}}$ and cubic sums ${S_{{l^3},l}}$ reduce to linear sums and polynomials in zeta values. The approach is based on constructive Power series and Cauchy product computations.

We study the Galois symbol map of the Milnor $K$-group attached to elliptic curves over a $p$-adic field. As by-products, we determine the structure of the Chow group for the product of elliptic curves over a $p$-adic field under some assumptions.

We use the theory of exponent pairs and Vaaler polynomials to show that any interval of the form $[x,x+x^{1/2}]$ contains an integral multiple $m^2r \in [x,x+x^{1/2}]$ of a perfect square $m^2$ with an integer $m > x^{0.281286}$.

We investigate a finite analogue of the Mordell-Tornheim multiple zeta values (the finite Mordell-Tornheim multiple zeta values). These values can be expressed by a linear combination of finite multiple zeta values, and its rules are described by the shuffle product. As a~corollary, we give a certain relation among finite multiple zeta values.

We study the multiplicative orders of $(\frac{n-1}{M})_n! \pmod{n}$ for odd prime powers $n=p^\alpha$, $p\equiv 1\pmod{M}$, where the Gauss factorial $N_n!$ denotes the product of all integers up to $N$ that are relatively prime to $n$. Departing from previously obtained results on the connection between the order for $p^\alpha$ and for $p^{\alpha+1}$, we obtain new criteria for exceptions to a general pattern, with particular emphasis on the cases $M=3$, $M=4$ and $M=6$. In the process we also obtain some results of independent interest. Most results are based on generalizations of binomial coefficient congruences of Gauss, Jacobi, and Hudson and Williams.

In the present paper, we study the approximation and growth of solutions to a class of elliptic partial differential equations. The characterizations of generalized order and generalized type of solutions to a class of elliptic partial differential equations have been obtained in terms of approximation errors.

For fixed $a\geq 2$, we suggest that the probability of nullity mod $p$ of the Fermat quotient $q_p(a)$ is $\ll \frac{1}{p}$ for $p \to \infty$. For this we propose various heuristics (as the existence of a suitable binomial law of probability), justified by means of numerical computations and analytical results, which may imply, via the Borel--Cantelli heuristic, that $q_p(a) \ne 0$ for all $p$ except a finite number (Th. 4.9). These heuristics are based on the possible existence (with an analogous probability) of $O(\log(p))$ ``abundant'' solutions $z_i \in [2, p-1[$ which are not necessarily of the ``exceptional'' form $a^k$, $1 \leq k < \log(p)/ \log(a)$, when $q_p(a)=0$, showing the exceptional solutions as a particular case of abundant solutions, for which a law of probability is natural. We also compute the density of integers $A$ such that $q_p(A) \ne 0, \forall p \leq x$ (Th. 4.12).