scholarly journals On the values of the Riemann zeta-function at rational arguments

2001 ◽  
Vol Volume 24 ◽  
Author(s):  
S Kanemitsu ◽  
Y Tanigawa ◽  
M Yoshimoto
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Companion Paper ◽  
Hurwitz Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Ramanujan’S Formula ◽  
Form Evaluation ◽  
Rational Arguments

International audience In a companion paper, ``On multi Hurwitz-zeta function values at rational arguments, Acta Arith. {\bf 107} (2003), 45-67'', we obtained a closed form evaluation of Ramanujan's type of the values of the (multiple) Hurwitz zeta-function at rational arguments (with denominator even and numerator odd), which was in turn a vast generalization of D. Klusch's and M. Katsurada's generalization of Ramanujan's formula. In this paper we shall continue our pursuit, specializing to the Riemann zeta-function, and obtain a closed form evaluation thereof at all rational arguments, with no restriction to the form of the rationals, in the critical strip. This is a complete generalization of the results of the aforementioned two authors. We shall obtain as a byproduct some curious identities among the Riemann zeta-values.

Null trajectories for the symmetrized Hurwitz zeta function

2006 ◽  
Vol 463 (2077) ◽  
pp. 303-319 ◽  
Author(s):  
Ross C McPhedran ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Hurwitz Zeta Function ◽  
Trigonometric Functions ◽  
Asymptotic Results ◽  
End Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Pass Through

We consider the Hurwitz zeta function ζ ( s , a ) and develop asymptotic results for a = p / q , with q large, and, in particular, for p / q tending to 1/2. We also study the properties of lines along which the symmetrized parts of ζ ( s , a ), ζ + ( s , a ) and ζ − ( s , a ) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of a =1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions L −3 ( s ) and L −4 ( s ), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of L −3 ( s ) and L −4 ( s ) to that of ζ ( s ) on the critical line.

scholarly journals QUESTIONS AROUND THE NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION. COMPUTATIONS AND CLASSIFICATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Ramūnas Garunkštis ◽  
Joern Steuding
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Hurwitz Zeta Function ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

We study the sequence of nontrivial zeros of the Riemann zeta-function with respect to sequences of zeros of other related functions, namely, the Hurwitz zeta-function and the derivative of Riemann's zeta-function. Finally, we investigate connections of the nontrivial zeros with the periodic zeta-function. On the basis of computation we derive several classifications of the nontrivial zeros of the Riemann zeta-function and stateproblems which mightbe ofinterestfor abetter understanding of the distribution of those zeros.

Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function

2020 ◽  
Vol 57 (2) ◽  
pp. 147-164
Author(s):  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Pair Correlation ◽  
Weak Form ◽  
Rational Numbers ◽  
Hurwitz Zeta Function ◽  
Linearly Independent ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Universality

AbstractLet 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.

scholarly journals Shifted quadratic Zeta series

2004 ◽  
Vol 2004 (67) ◽  
pp. 3631-3652
Author(s):  
Anthony Sofo
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Point Of View ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is well known that the Riemann Zeta functionς(p)=∑n=1∞1/npcan be represented in closed form forpan even integer. We will define a shifted quadratic Zeta series as∑n=1∞1/(4n2−α2)p. In this paper, we will determine closed-form representations of shifted quadratic Zeta series from a recursion point of view using the Riemann Zeta function. We will also determine closed-form representations of alternating sign shifted quadratic Zeta series.

scholarly journals Ramanujan's formula for the logarithmic derivative of the gamma function

1996 ◽  
Vol 120 (3) ◽  
pp. 391-401
Author(s):  
David Bradley
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Logarithmic Derivative ◽  
Euler’S Constant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Ramanujan’S Formula ◽  
Positive Integers ◽  
Euler's Constant

AbstractWe prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

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