An Optimization Problem Related to the Zeta-function

1986 ◽  
Vol 29 (1) ◽  
pp. 70-73 ◽  
Author(s):  
Silviu Guiasu
Keyword(s):  
Probability Distribution ◽  
Zeta Function ◽  
Unique Solution ◽  
Optimization Problem ◽  
Maximization Problem ◽  
Entropy Maximization ◽  
Riemann’S Zeta Function ◽  
Positive Integers

AbstractS. Golomb noticed that Riemann's zeta function ζ induces a probability distribution on the positive integers, for any s > 1, and studied some of its properties connected to divisibility. The object of this paper is to show that the probability distribution mentioned above is the unique solution of an entropy-maximization problem.

scholarly journals The Sums of Alternating Series of Odd Powers of the Reciprocals of Odd Positive Integers

2017 ◽  
Vol 1 (1) ◽  
pp. 18-26
Author(s):  
Mohammed R. Karim
Keyword(s):  
Zeta Function ◽  
Recent Work ◽  
Higher Power ◽  
Riemann’S Zeta Function ◽  
Work Done ◽  
Positive Integers

This paper is an extension of a recent work done by the author [4] and here the sums of alternating series of odd powers (up to fifteen) of the reciprocals of odd positive integers are computed. Following this method, the sum of the series of any higher power could be calculated. In the process of computing these sums, the sums of the series of even powers of reciprocals of odd positive integers have been reestablished and enabled the author to compute the values of Riemann’s zeta function for even positive integers.

On multiple Hurwitz zeta function of Mordell–Tornheim type

2021 ◽  
pp. 1-34
Author(s):  
Kazuhiro Onodera
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Multiple Zeta Function ◽  
Multiple Zeta ◽  
Second Derivatives ◽  
Positive Integers

We introduce a certain multiple Hurwitz zeta function as a generalization of the Mordell–Tornheim multiple zeta function, and study its analytic properties. In particular, we evaluate the values of the function and its first and second derivatives at non-positive integers.

THE SHUFFLE VARIANT OF JEŚMANOWICZ’ CONJECTURE CONCERNING PYTHAGOREAN TRIPLES

2011 ◽  
Vol 90 (3) ◽  
pp. 355-370
Author(s):  
TAKAFUMI MIYAZAKI
Keyword(s):  
Unique Solution ◽  
Pythagorean Triples ◽  
Pythagorean Triple ◽  
Positive Integers

AbstractLet (a,b,c) be a primitive Pythagorean triple such that b is even. In 1956, Jeśmanowicz conjectured that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in the positive integers. This is one of the most famous unsolved problems on Pythagorean triples. In this paper we propose a similar problem (which we call the shuffle variant of Jeśmanowicz’ problem). Our problem states that the equation cx+by=az with x,y and z positive integers has the unique solution (x,y,z)=(1,1,2) if c=b+1 and has no solutions if c>b+1 . We prove that the shuffle variant of the Jeśmanowicz problem is true if c≡1 mod b.

scholarly journals Entropy Balancing is Doubly Robust

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Qingyuan Zhao ◽  
Daniel Percival
Keyword(s):  
Propensity Score ◽  
Observational Studies ◽  
Optimization Problem ◽  
Entropy Maximization ◽  
Entropy Balancing ◽  
Covariate Balance ◽  
Original Proposal ◽  
Variance Bound ◽  
Doubly Robust ◽  
Theoretical Results

AbstractCovariate balance is a conventional key diagnostic for methods estimating causal effects from observational studies. Recently, there is an emerging interest in directly incorporating covariate balance in the estimation. We study a recently proposed entropy maximization method called Entropy Balancing (EB), which exactly matches the covariate moments for the different experimental groups in its optimization problem. We show EB is doubly robust with respect to linear outcome regression and logistic propensity score regression, and it reaches the asymptotic semiparametric variance bound when both regressions are correctly specified. This is surprising to us because there is no attempt to model the outcome or the treatment assignment in the original proposal of EB. Our theoretical results and simulations suggest that EB is a very appealing alternative to the conventional weighting estimators that estimate the propensity score by maximum likelihood.

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