Related papers: Conditional upper bound for the k-th prime ideal w…

We prove an effective version of the Chebotarev theorem for the density of prime ideals with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions.

Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $\zeta_K(s)$ and used this to give effective upper and lower bounds on the…

Let $\zeta_K(s)$ denote the Dedekind zeta-function associated to a number field $K$. In this paper, we give an effective upper bound for the height of first non-trivial zero other than $1/2$ of $\zeta_K(s)$ under the generalized Riemann…

Let $\psi_\K$ be the Chebyshev function of a number field $\K$. Under GRH we prove an explicit upper bound for $|\psi_\K(x)-x|$ in terms of the degree and the discriminant of $\K$. The new bound improves significantly on previous known…

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$ for every positive integer k. Our bounds are nearly as sharp as…

Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in $[T-a,T+a]$. We also prove a bound for the multiplicity of the zeros.

We have proved recently several explicit versions of the prime ideal theorem under GRH. Here we prove a version with optimal asymptotic behaviour.

We find an explicit upper bound for general $L$-functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of $L$-functions and Dedekind zeta functions.…

Assuming the Riemann Hypothesis, we provide effective upper and lower estimates for $\left|\zeta(s)\right|$ right to the critical line. As an application we make explicit Titchmarsh's conditional bound for the Mertens function and…

Assuming the Riemann hypothesis, we establish an upper bound for the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where $k$ is a positive real number. Our upper bound agrees with conjectures of…

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

In this short note, we give a proof of the Riemann hypothesis for Goss $v$-adic zeta function $\zeta_{v}(s)$, when $v$ is a prime of $\mathbb{F}_{q}[t]$ of degree one.

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group…

We prove explicit upper bounds of the function $S_m(T)$, defined by the repeated integration of the argument of the Riemann zeta-function. The explicit upper bound of $S(T)$ and $S_1(T)$ have already been obtained by A. Fujii. Our result is…

In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $\zeta(s)$ valid for $\Re(s)>1$, as well as similar formulas for $\zeta(s)$ valid for an even and odd $k$th positive integer argument.…

Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet $L$-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen…

Let $L$ be a finite Galois extension of the number field $K$. We unconditionally bound the least prime ideal of $K$ occurring in the Chebotarev Density Theorem as a power of the discriminant of $L$ with an explicit exponent. We also…

We prove an explicit upper bound of the function $S(t,\chi)$, defined by the argument of Dirichlet $L$-functions. An explicit upper bound of the function $S_1(t)$, defined by the integral of the argument of the Riemann zeta-function, have…

In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the true upper and lower bound of the residues of Dedekind zeta functions except for a density zero set. For $S_5$-fields, we need to assume the strong Artin conjecture. We also show…

We introduce Riemann operators acting on Quillen's higher $K$--groups $K_{n}(A)$ for the integer ring $A$ of an algebraic number field $K$. Especially we prove that gamma factors of Dedekind zeta function of $K$ are obtained as regularized…