scholarly journals An unexpected property of odd order derivatives of Hardy’s function

2014 ◽  
Vol 95 (109) ◽  
pp. 173-188 ◽  
Author(s):  
Philippe Blanc
Keyword(s):  
Riemann Hypothesis ◽  
Odd Order ◽  
Hardy’S Function ◽  
Derivatives Of ◽  
Unexpected Behavior

Assuming the Riemann hypothesis, we show that the odd order derivatives of Hardy?s function have, under some condition, an unexpected behavior for large values of t.

scholarly journals Properties of extrema of estimates for middle derivatives of odd order in Sobolev classes

2019 ◽  
Vol 487 (5) ◽  
pp. 487-492
Author(s):  
T. A. Garmanova ◽  
I. A. Sheipak
Keyword(s):  
Sobolev Spaces ◽  
Global Maximum ◽  
Sobolev Classes ◽  
Odd Order ◽  
Derivatives Of

The embedding constants for the Sobolev spaces W2n[0;1]→W∞k[0; 1], 0 ≤ k ≤ n - 1 are considered. The properties of the functions An,k(x) arising in the inequalities |f(k)(x)|≤An,k (x)││f||W2n[0;1], are studied. The extremum points of An;k are calculated for k = 3, 5 and all admissible n. The global maximum of these functions is found, and the exact embedding constants are calculated.

On the distribution of zeros of derivatives of the Riemann ξ-function

2020 ◽  
Vol 32 (1) ◽  
pp. 1-22
Author(s):  
Amita Malik ◽  
Arindam Roy
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Real Number ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Density Estimate ◽  
Distribution Of Zeros ◽  
Riemann Zeta ◽  
Derivatives Of

AbstractFor the completed Riemann zeta function {\xi(s)}, it is known that the Riemann hypothesis for {\xi(s)} implies the Riemann hypothesis for {\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence {(\alpha\gamma_{m})}, where α is any fixed non-zero real number and {\gamma_{m}} runs over the imaginary parts of the zeros of {\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of {\xi^{(m)}(s)}. In particular, all our results hold uniformly for {0\leq m\leq g(T)}, where the function {g(T)} tends to infinity with T and {g(T)=o(\log\log T)}.

Performance and Complexity Evaluation of OTR-UWB Receiver

2011 ◽  
pp. 168-181
Author(s):  
Hossein Gharaee ◽  
Abdolreza Nabavi ◽  
Jalil ("Joe") Etminan
Keyword(s):  
Hardware Complexity ◽  
Odd Order ◽  
Data Rates ◽  
Quartus Ii ◽  
Orthogonal Property ◽  
Simulation Results ◽  
Derivatives Of ◽  
New System ◽  
Ber Performance ◽  
Uwb Receiver

This article presents a new transmitted reference UWB receiver, which utilizes the orthogonal property of even and odd order derivatives of Gaussian pulses in neighboring chips for synchronization. This system, referred to as orthogonal TR-UWB (OTR-UWB), employs only a single spreading code, which results in much lower mean detection time compared to DS-UWB systems. The hardware complexity for OTR-UWB receiver is significantly reduced against conventional TR-UWB systems. In addition, simulation results show that BER performance is improved, while the new system is capable of supporting higher data rates. Also, this article presents the FPGA implementation of OTR-UWB, with a bit-rate of 25Mb/s without using equalizer. In addition, we present the DSP algorithm of baseband. Hardware of this system is implemented on two different FPGAs from ALTERA and XILINX, CycloneII (EP2C35F672C6) and Spartan 3 (3s4000fg676-5). Gate estimation and power analysis are performed by Quartus II 7.2 (ALTERA) and ISE 8.1 (XILINX) softwares. [Article copies are available for purchase from InfoSci-on-Demand.com]

Biased behavior of weighted Mertens sums

2019 ◽  
Vol 16 (03) ◽  
pp. 547-577
Author(s):  
Emre Alkan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Prime Numbers ◽  
Summation Formulas ◽  
The Riemann Zeta Function ◽  
Convexity Properties ◽  
Riemann Zeta ◽  
Derivatives Of

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at [Formula: see text].

scholarly journals Logarithmic derivatives of Artin -functions

2013 ◽  
Vol 149 (4) ◽  
pp. 568-586 ◽  
Author(s):  
Peter J. Cho ◽  
Henry H. Kim
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Number Fields ◽  
Upper And Lower Bounds ◽  
Zero Density ◽  
Galois Closures ◽  
Derivatives Of ◽  
Logarithmic Derivatives

AbstractLet $K$ be a number field of degree $n$, and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$-functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$, respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$, $D_n$ for $n=3,4,5$, $S_4$ or $A_5$.

Upper bounds for the moments of derivatives of Dirichlet L-functions

2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Keiju Sono
Keyword(s):  
Riemann Hypothesis ◽  
Upper Bounds ◽  
Generalized Riemann Hypothesis ◽  
Derivatives Of

AbstractIn this paper, we give certain upper bounds for the 2k-th moments, k ≥ 1/2, of derivatives of Dirichlet L-functions at s = 1/2 under the assumption of the Generalized Riemann Hypothesis.

Performance and Complexity Evaluation of OTR-UWB Receiver

2010 ◽  
pp. 1945-1958
Author(s):  
Hossein Gharaee ◽  
Abdolreza Nabavi ◽  
Jalil ("Joe") Etminan
Keyword(s):  
Hardware Complexity ◽  
Odd Order ◽  
Data Rates ◽  
Quartus Ii ◽  
Orthogonal Property ◽  
Simulation Results ◽  
Derivatives Of ◽  
New System ◽  
Ber Performance ◽  
Uwb Receiver

This article presents a new transmitted reference UWB receiver, which utilizes the orthogonal property of even and odd order derivatives of Gaussian pulses in neighboring chips for synchronization. This system, referred to as orthogonal TR-UWB (OTR-UWB), employs only a single spreading code, which results in much lower mean detection time compared to DS-UWB systems. The hardware complexity for OTR-UWB receiver is significantly reduced against conventional TR-UWB systems. In addition, simulation results show that BER performance is improved, while the new system is capable of supporting higher data rates. Also, this article presents the FPGA implementation of OTR-UWB, with a bit-rate of 25Mb/s without using equalizer. In addition, we present the DSP algorithm of baseband. Hardware of this system is implemented on two different FPGAs from ALTERA and XILINX, CycloneII (EP2C35F672C6) and Spartan 3 (3s4000fg676-5). Gate estimation and power analysis are performed by Quartus II 7.2 (ALTERA) and ISE 8.1 (XILINX) softwares.

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