We study the zero-density estimates for automorphic
L
-functions
L
(
s
,
π
)
for
GL
m
when
σ
is near 1. In particular, we get a range of
σ
for which the density hypothesis holds. The proofs use a ...zero detecting argument, the Halász–Montgomery inequality and a bound for an integral power moment of
L
(
1
/
2
+
i
t
,
π
)
.
Investigation has been made regarding the properties of the ℿ
(1
1/p
) products over the prime numbers, where we fix the s
ℝ exponent, and let the n
2 natural bound grow toward positive infinity. The ...nature of these products for the s
1 case is known. We get approximations for the case when s
1/2, 1), furthermore different observations for the case when s<1/2.
The Newton power-sum formulas relate to sums of powers of roots of a polynomial with the coefficients of the polynomial. In this paper we obtain formulas that relate to sums of reciprocal powers of ...zeros and poles of entire and meromorphic functions with the coefficients of their Taylor series expansions. We then derive the recurrence formulas for the Riemann zeta function at integer arguments and compute the sums extended over the nontrivial zeros of the Riemann zeta function.
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.
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 .
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.
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