Comment on the sums Silagadze, Zurab
Georgian mathematical journal,
09/2012, Volume:
19, Issue:
3
Journal Article
Peer reviewed
Open access
This is a comment on the papers from Elkies Amer. Math. Monthly 110 (2003), 561–573 and Cvijović–Klinowski J. Comput. Appl. Math. 142 (2002), 435–439. We provide an explicit expression for the kernel ...of the integral operator introduced in Elkies' paper. This explicit expression considerably simplifies the calculation of and enables a simple derivation of Cvijović and Klinowski's integral representation for .
The author’s research devoted to the Hilbert’s double series theorem and its various further extensions are the focus of a recent survey article. The sharp version of double series inequality result ...is extended in the case of a not exhaustively investigated non-homogeneous kernel, which mutually covers the homogeneous kernel cases as well. Particularly, novel Hilbert’s double series inequality results are presented, which include the upper bounds built exclusively with non-weighted ℓp–norms. The main mathematical tools are the integral expression of Mathieu (a,λ)-series, the Hölder inequality and a generalization of the double series theorem by Yang.
We investigate the second moment of a random sampling ζ(1/2+iXt) of the Riemann zeta function on the critical line. Our main result states that if Xt is an increasing random sampling with gamma ...distribution, then for all sufficiently large t,E|ζ(1/2+iXt)|2=logt+O(logtloglogt).
In this paper the fractional order derivative of a Dirichlet series, Hurwitz zeta function and Riemann zeta function is explicitly computed using the Caputo fractional derivative in the Ortigueira ...sense. It is observed that the obtained results are a natural generalization of the integer order derivative. Some interesting properties of the fractional derivative of the Riemann zeta function are also investigated to show that there is a chaotic decay to zero (in the Gaussian plane) and a promising expression as a complex power series.
Essential bounds of Dirichlet polynomials Mora, G.; Benítez, E.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas,
07/2021, Volume:
115, Issue:
3
Journal Article
Peer reviewed
Open access
In this paper we have given conditions on exponential polynomials
P
n
(
s
)
of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds,
a
P
n
(
s
)
,
ρ
...N
and
b
P
n
(
s
)
,
ρ
0
associated with
P
n
(
s
)
. The reciprocal question has been also treated. The bounds
a
P
n
(
s
)
,
b
P
n
(
s
)
are defined as the end-points of the minimal closed and bounded real interval
I
=
a
P
n
(
s
)
,
b
P
n
(
s
)
such that all the zeros of
P
n
(
s
)
are contained in the strip
I
×
R
of the complex plane
C
. The bounds
ρ
N
,
ρ
0
are defined as the unique real solutions of Henry equations of
P
n
(
s
)
. Some applications to the partial sums of the Riemann zeta function have been also showed.
In this paper, the following multiple fractional part integrals In,βα1,α2,⋯,αn=∫0,1n∏j=1nxjαj{Sn−1}βdx1⋯dxn and Jn,βα=∫0,1nSnα{Sn−1}βdx1⋯dxn are studied for positive integer n and complex values of ...α,β,αj(j=1,2,⋯,n), where {u} denotes the fractional part of u, R(s) denotes the real part of s and Sn=x1+x2+⋯+xn. It is proved that I1,βα can be represented as a linear combination of the Riemann zeta function, the Beta function and Euler’s constant as R(β)>−1. Moreover, In,βα1,α2,⋯,αn can be expressed by In−1,βα1,α2,⋯,αn−1, the Beta function and the incomplete Beta function for n=2,3. In addition, the recurrence formula of Jn,βα(n=2,3,⋯) is established and Jn,βα can be expressed by I1,βα, logarithmic function and some binomial coefficients.
We introduce a family of heavy-traffic regimes for large-scale service systems, presenting a range of scalings that include both moderate and extreme heavy traffic, as compared to classical heavy ...traffic. The heavy-traffic regimes can be translated into capacity sizing rules that lead to economies-of-scales, so that the system utilization approaches 100% while congestion remains limited. We obtain heavy-traffic approximations for stationary performance measures in terms of asymptotic expansions, using a nonstandard saddle point method, tailored to the specific form of integral expressions for the performance measures, in combination with the heavy-traffic regimes.
NOTES ON log(ζ(𝑠)) STOPPLE, JEFFREY
The Rocky Mountain journal of mathematics,
01/2016, Volume:
46, Issue:
5
Journal Article
Peer reviewed
Open access
Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann 𝜁 function, in particular, the zeros of this function. ...Theorem 1.2 gives a zero-free region. Theorem 1.4 gives an asymptotic estimate for the number of nontrivial zeros to height 𝑇. Theorem 1.7 is a zero density estimate.
Inspired by representations of the class number of imaginary quadratic fields, in this paper, we give explicit evaluations of trigonometric series having generalized harmonic numbers as coefficients ...in terms of odd values of the Riemann zeta function and special values of L-functions subject to the parity obstruction. The coefficients that arise in these evaluations are shown to belong to certain cyclotomic extensions. Furthermore, using best polynomial approximation of smooth functions under uniform convergence due to Jackson and their log-sine integrals, we provide approximations of real numbers by combinations of special values of L-functions corresponding to the Legendre symbol. Our method for obtaining these results rests on a careful study of generating functions on the unit circle involving generalized harmonic numbers and the Legendre symbol, thereby relating them to values of polylogarithms and then finally extracting Fourier series of special functions that can be expressed in terms of Clausen functions.
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