We introduce poly-Euler polynomials, which generalize Euler polynomials. Various results about them are provided. Furthermore, we introduce zeta functions of Arakawa--Kaneko type, and discuss their properties and the relation with poly-Euler polynomials.

Generalizing work of Polya, de Bruijn and Newman, we allow the backward heat equation to deform the zeros of quadratic Dirichlet $L$-functions. There is a real constant $\Lambda_{Kr}$ (generalizing the de Bruijn-Newman constant $\Lambda$) such that for time $t\ge\Lambda_{Kr}$ all such $L$-functions have all their zeros on the critical line; for time $t<\Lambda_{Kr}$ there exist zeros off the line. Under GRH, $\Lambda_{Kr}\le 0$; we make the complementary conjecture $0\le \Lambda_{Kr}$. Following the work of Csordas \emph{et. al}. on Lehmer pairs of Riemann zeros, we use low-lying zeros of quadratic Dirichlet $L$-functions to show that $-1.13\cdot 10^{-7}<\Lambda_{Kr}$. In the last section we develop a precise definition of a Low discriminant which is motivated by considerations of random matrix theory. The existence of infinitely many Low discriminants would imply $0\le \Lambda_{Kr}$.

We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-function of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new formulas which especially include the parity results of double and triple zeta values. As another important application, we give certain refinement of restricted sum formulas, which gives restricted sum formulas among MZVs of an arbitrary depth $r$ which were previously known only in the cases of depth $2,3,4$. Furthermore, considering a~sub-root system of type $B_r$ analogously, we can give relevant analogues of the Hoffman-Zagier formula, parity results and restricted sum formulas.

Explicit congruences (mod $p$) are proved for the class equations corresponding to discriminants $D=-8p, -3p, -12p$ in the theory of complex multiplication, where $p$ is an odd prime. They are explicit in that they can be computed directly from a formula for the supersingular polynomial without first having to know the coefficients of the class equation in characteristic zero. These congruences have previously appeared in print without proof, and have been used to study factorizations of certain Legendre polynomials (mod $p$).

The famous sum formula of multiple zeta values (MZV) says that the sum of all MZVs of fixed weight w and depth d is always equal to (w). Hoffman proved a more complicated formula when all the arguments of the MZVs are even numbers. In this paper, we further restrict the arguments to multiples of $4$ and derive a similar sum formula.

This paper is to show a non-vanishing property of the derivatives of certain class of $L$-functions. We study the non-vanishing and transcendence of special values of $L$-functions and their derivatives, attached to (cuspidal) Siegel-Hecke eigenforms of genus $2$, quadratic twists of classical Hecke eigenforms, and half-integral weight modular forms.

Let $B$ be a compact operator in a Hilbert space $H$ and $S$ an unbounded normal one in $H$, having a compact resolvent. We consider operators of the form $A=S+B$. Numerous integro-differential operators $A$ can be represented in this form. The paper deals with approximations of the eigenvalues of the considered operators by the eigenvalues of the operators $A_n=S+B_n$ $(n=1,2,...)$, where $B_n$ are $n$-dimensional operators. Besides, we obtain the error estimate of the approximation.

There are many classes of ill-posed problems that cannot be solved with existing iterative methods, since the usual Lipschitz-type assumptions are not satisfied. In this study, we expand the applicability of a two step Newton-type projection method considered in [10,11], using weaker assumptions. Numerical examples for the method and examples where the old assumptions are not satisfied but the new assumptions are satisfied are provided at the end of this study.

In the preceding papers, we studied the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{p})$ for an odd prime number $p$ using certain units and the invariants $n_0^{(r)}$ and $n_2$. In the present paper, we develop new criteria for Greenberg conjecture using $n_0^{(r)}$ and $n_2$.

We are looking for real numbers $\alpha$ and $d$ for which there exist ``many'' real numbers $\tau$ such that the shifts of the Hurwitz-zeta function $\zeta(s+i\tau,\alpha)$ and $\zeta(s+id\tau,\alpha)$ are ``near'' each other.

Lehmer's totient problem asks whether there exists a composite $n$ such that $\varphi(n)\mid (n-1)$, where $\varphi$ is the Euler's function. This problem is still open. Later, several upper bounds of the derived problem ``$\varphi(n)\mid (n-a)$'' were given. In this paper, we extend it to $n$ with $\varphi(n) \mid (\kappa n-a)$ and obtain some new bounds. As an application, for any integer $\lambda>0$ we have, \[ \#\{n\leq x: \varphi(n)\mid (n-1), n\not \equiv 1 (mod 6^{\lambda})\}\ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)},\\ \#\{n\leq x: \varphi(n)\mid (3n-1)\} \ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)}. \]

In this paper we show discrepancy bounds for index-transformed uniformly distributed sequences. From a general result we deduce very tight lower and upper bounds on the discrepancy of index-transformed van der Corput-, Halton-, and $(t,s)$-sequences indexed by the sum-of-digits function. We also analyze the discrepancy of sequences indexed by other functions, such as, e.g., $\lfloor n^{\alpha}\rfloor$ with $0 < \alpha < 1$.