Non-vanishing of Dirichlet series without Euler products

International Audience ◽

Euler Products ◽

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

Notes on the Riemann zeta Function-III

10.46298/hrj.1999.139 ◽

1999 ◽

Vol Volume 22 ◽

Author(s):

R Balasubramanian ◽

K Ramachandra ◽

A Sankaranarayanan ◽

K Srinivas

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

International Audience ◽

International audience For a good Dirichlet series $F(s)$ (see Definition in \S1) which is a quotient of some products of the translates of the Riemann zeta-function, we prove that there are infinitely many poles $p_1+ip_2$ in $\Im (s)>C$ for every fixed $C>0$. Also, we study the gaps between the ordinates of the consecutive poles of $F(s)$.

A Lemma in complex function theory I

10.46298/hrj.1989.108 ◽

1989 ◽

Vol Volume 12 ◽

Author(s):

R Balasubramanian ◽

K Ramachandra

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Function Theory ◽

Complex Function ◽

Complex Function Theory ◽

Closed Disc ◽

International Audience ◽

International audience Continuing our earlier work on the same topic published in the same journal last year we prove the following result in this paper: If $f(z)$ is analytic in the closed disc $\vert z\vert\leq r$ where $\vert f(z)\vert\leq M$ holds, and $A\geq1$, then $\vert f(0)\vert\leq(24A\log M) (\frac{1}{2r}\int_{-r}^r \vert f(iy)\vert\,dy)+M^{-A}.$ Proof uses an averaging technique involving the use of the exponential function and has many applications to Dirichlet series and the Riemann zeta function.

Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1

10.46298/hrj.1978.87 ◽

1978 ◽

Vol Volume 1 ◽

Author(s):

K Ramachandra

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Positive Constant ◽

Riemann Zeta Function ◽

Nonnegative Integer ◽

Mean Value ◽

International Audience ◽

Riemann Zeta ◽

The Mean

International audience The present paper is concerned with $\Omega$-estimates of the quantity $$(1/H)\int_{T}^{T+H}\vert(d^m/ds^m)\zeta^k(\frac{1}{2}+it)\vert dt$$ where $k$ is a positive number (not necessarily an integer), $m$ a nonnegative integer, and $(\log T)^{\delta}\leq H \leq T$, where $\delta$ is a small positive constant. The main theorems are stated for Dirichlet series satisfying certain conditions and the corollaries concerning the zeta function illustrate quite well the scope and interest of the results. %It is proved that if $2k\geq1$ and $T\geq T_0(\delta)$, then $$(1/H)\int_{T}^{T+H}\vert \zeta(\frac{1}{2}+it)\vert^{2k}dt > (\log H)^{k^2}(\log\log H)^{-C}$$ and $$(1/H)\int_{T}^{T+H} \vert\zeta'(\frac{1}{2}+it)\vert dt > (\log H)^{5/4}(\log\log H)^{-C},$$ where $C$ is a constant depending only on $\delta$.

The convergence of Euler products over p-adic number fields

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091507000636 ◽

2009 ◽

Vol 52 (3) ◽

pp. 583-606 ◽

Cited By ~ 1

Author(s):

Daniel Delbourgo

Keyword(s):

Topological Space ◽

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Number Fields ◽

Product Structure ◽

Euler Product ◽

Euler Products ◽

Riemann Zeta ◽

New Formula

AbstractWe define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

Alternating Euler sums at the negative integers.

10.46298/hrj.2009.165 ◽

2009 ◽

Vol Volume 32 ◽

Author(s):

Khristo Boyadzhiev ◽

H. Gopalkrishna Gadiyar ◽

R Padma

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Complex Plane ◽

Riemann Zeta Function ◽

Euler Numbers ◽

Euler Sums ◽

Entire Complex ◽

International Audience ◽

International audience We study three special Dirichlet series, two of them alternating, related to the Riemann zeta-function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.

A RELATION BETWEEN THE ZEROS OF TWO DIFFERENT L-FUNCTIONS WHICH HAVE AN EULER PRODUCT AND FUNCTIONAL EQUATION

International Journal of Number Theory ◽

10.1142/s1793042105000248 ◽

2005 ◽

Vol 01 (03) ◽

pp. 401-429

Author(s):

MASATOSHI SUZUKI

Keyword(s):

Functional Equation ◽

Zeta Function ◽

Riemann Zeta Function ◽

Modular Forms ◽

Functional Equations ◽

Euler Product ◽

Euler Products ◽

Riemann Zeta ◽

Special Case

As automorphic L-functions or Artin L-functions, several classes of L-functions have Euler products and functional equations. In this paper we study the zeros of L-functions which have Euler products and functional equations. We show that there exists a relation between the zeros of the Riemann zeta-function and the zeros of such L-functions. As a special case of our results, we find relations between the zeros of the Riemann zeta-function and the zeros of automorphic L-functions attached to elliptic modular forms or the zeros of Rankin–Selberg L-functions attached to two elliptic modular forms.

Notes on the Riemann zeta-function-IV

10.46298/hrj.1999.140 ◽

1999 ◽

Vol Volume 22 ◽

Author(s):

R Balasubramanian ◽

K Ramachandra ◽

A Sankaranarayanan

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

International Audience ◽

International audience For ``good Dirichlet series'' $F(s)$ we prove that there are infinitely many poles $p_1+ip_2$ in $\Im (s)>C$ for every fixed $C>0$. Also we study the gaps between the numbers $p_2$ arranged in the non-decreasing order.

Dirichlet series induced by the Riemann zeta-function

Studia Mathematica ◽

10.4064/sm187-2-4 ◽

2008 ◽

Vol 187 (2) ◽

pp. 157-184

Author(s):

Jun-ichi Tanaka

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Dirichlet series related to the Riemann zeta function

Journal of Number Theory ◽

10.1016/0022-314x(84)90094-5 ◽

1984 ◽

Vol 19 (1) ◽

pp. 85-102 ◽

Cited By ~ 23

Author(s):

Tom M. Apostol ◽

Thiennu H. Vu

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

International Journal of Number Theory ◽

10.1142/s1793042118500252 ◽

2018 ◽

Vol 14 (02) ◽

pp. 371-382

Author(s):

K. Paolina Koutsaki ◽

Albert Tamazyan ◽

Alexandru Zaharescu

Keyword(s):

Asymptotic Formula ◽

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Linear Combinations ◽

The Real ◽

Unique Integer ◽

Riemann Zeta ◽

Derivatives Of

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.