Related papers: Change of variable in Riemann integration revisite…
A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.
In this note, we prove Selberg's announced result on $r$-gaps between zeros of the Riemann zeta-function $\zeta$. Our proof uses a result on variations of $\arg\zeta$ by Tsang based on Selberg's method. The same result with explicit…
Integrating with respect to functions which are constant on intervals whose bounds are discontinuity points (of those functions) is frequent in many branches of Mathematics, specially in stochastic processes. For such functions and alike…
We prove Ehrhard's inequality using interpolation along the Ornstein-Uhlenbeck semi-group. We also provide an improved Jensen inequality for Gaussian variables that might be of independent interest.
We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.
This paper has been withdrawn by the author, due to a crucial error in page 5.
We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work [3].
This paper presents new sufficient and equivalent conditions for the generalized version of the Riemann Hypothesis. The paper derives also statements and remarks concerning zero-free regions, modified Hadamard-product formulas and the…
In this paper, we prove a generalization of Reilly's formula in \cite{Reilly}. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the hemisphere and in the…
Leggett formulated an inequality which seems to generalize the Bell theorem to non-local hidden variable theories. Leggett inequality is violated by quantum mechanics, as was confirmed by experiment. However, a careful analysis reveals that…
We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.
The paper presents two arithmetical versions of the Nyman-Beurling equivalence with the Riemann hypothesis, proved by classical, quasi elementary, number-theoretic methods, based on an integrated version of the classical combinatorial…
We discuss the application of the determinantal method to the proof of the Riemann hypothesis. We start from the fact that, if a certain doubly infinite set of determinants are all positive, then the hypothesis is true. This approach…
In this note, we extend Euler's transformation formula from the alternating series to more general series. Then we give new expressions for the Riemann zeta function $\zeta(s)$ by the generalized difference operator $\Delta_{c}$, which…
Assuming the validity of Riemann Hypothesis (RH), we derive the explicit bilateral estimates ("narrow passage") of the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals. These results are reversable,…
We find it absurd that Walliser [1] essentially used the same analysis and obtained identical results as reported in [3], yet arrived at different conclusions. Namely, based on an incomplete theory and using erroneous arguments, he not only…
In this paper we go on to discuss about Stanley's theorem in Integer partitions. We give two different versions for the proof of the generalization of Stanley's theorem illustrating different techniques that may be applied to profitably…
A quite fast proof of the functional equation of the Riemann zeta function. It is a modification of a proof usually overlooked in Titchmarsh's monograph.
We present a new approach to proving non-termination of non-deterministic integer programs. Our technique is rather simple but efficient. It relies on a purely syntactic reversal of the program's transition system followed by a…
The aim of this paper is to give a proof of improving of Zalcman's lemma.