Proof of the Euler product formula for the Riemann zeta function
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Evaluation of Euler-like sums via Hurwitz zeta values Open
In this paper we collect two generalizations of harmonic numbers (namelygeneralized harmonic numbers and hyperharmonic numbers) under one roof.Recursion relations, closed-form evaluations, and generating functions of thisunified extension …
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Computation of the Riemann Zeta Function for deriving the Euler Product Open
The Euler product is an infinite product of many functions in the multiplicative number theory. A completely multiplicative function gives the Euler product which is equal to the Riemann zeta function. This paper presents the computation o…
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Representation of the Euler Product for the Riemann Zeta Function Open
The Riemann zeta function is the important special function in the large family of zeta functions. The Euler product is an infinite product of many functions in the multiplicative number theory. A completely multiplicative function gives t…
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On the logarithm of the Riemann zeta-function and its iterated integrals Open
This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the …
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Disproof of the Euler Product equal to the Riemann Zeta Function Open
The Euler product is equal to the Riemann zeta function: it was proved by the earlier mathematician. The author of this article shows that the Euler product is not equal to the Riemann zeta function.
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The Riemann zeta function and classes of infinite series Open
We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(n) n sn and we apply our results to obtain new representations for some mathematical cons…
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Shifted moments of the Riemann zeta function Open
In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$ th moments of the Riema…
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The inverses of tails of the Riemann zeta function Open
We present some bounds of the inverses of tails of the Riemann zeta function on and compute the integer parts of the inverses of tails of the Riemann zeta function for , and .
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New $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomials and their matrices Open
In this paper, we introduce the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic…
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Euler sums of generalized harmonic numbers and connected extensions Open
This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers H(p,q)n ?H(p,q)(r) = ?Xn=1 H(p,q)n/nr in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose ter…
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Explicit evaluations for several variants of Euler sums Open
We study several variants of Euler sums by using the methods of contour integration and the residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also defi…
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Evaluations of nonlinear Euler sums of weight ten Open
In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all quadr…
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The Riemann Hypothesis-Millennium Prize Problem Open
This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied; these properties lead to new regularities of zeta functions. The choice of a special type of m…
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On harmonic numbers and nonlinear Euler sums Open
In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta functi…
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Two Variants of Euler Sums Open
For positive integers $p_1,p_2,\ldots,p_k,q$ with $q>1$, we define the Euler $T$-sum $T_{p_1p_2\cdots p_k,q}$ as the sum of those terms of the usual infinite series for the classical Euler sum $S_{p_1p_2\cdots p_k,q}$ with odd denominators…
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Euler sums and non-integerness of harmonic type sums Open
We show that Euler sums of generalized hyperharmonic numbers can be evaluated in terms of Euler sums of generalized harmonic numbers and special values of the Riemann zeta function. Then we focus on the non-integerness of generalized hyper…
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Some series involving the Euler zeta function Open
In this paper, using the Boole summation formula, we obtain a new integral representation of $n$-th quasi-periodic Euler functions $\\overline{E}_n(x)$ for $n=1,2,\\ldots.$ We also prove several series involving Euler zeta functions $\\zet…
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Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$ Open
We establish lower bounds for the discrete 2 k th moment of the derivative of the Riemann zeta function at nontrivial zeros for all under the Riemann hypothesis and the assumption that all zeros of are simple.
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An investigation of the non-trivial zeros of the Riemann zeta function Open
While many zeros of the Riemann zeta function are located on the critical line $\Re(s)=1/2$, the non-existence of zeros in the remaining part of the critical strip $\Re(s) \in \, ]0, 1[$ is the main scope to be proven for the Riemann hypot…
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Zeta functions interpolating the convolution of the Bernoulli polynomials Open
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Note on the Riemann Hypothesis: International Conference on Recent Developments in Mathematics (ICRDM 2022) Open
The Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. In 1984, Guy Robin stated a new criterion for the Riemann Hypothesis. We prove t…
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ON THE (p, q)-ANALOGUE OF EULER ZETA FUNCTION Open
In this paper we define (p, q)-analogue of Euler zeta function. In order to define (p, q)-analogue of Euler zeta function, we introduce the (p, q)-analogue of Euler numbers and polynomials by generalizing the Euler numbers and polynomials,…
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New infinite product formulas for the Riemann zeta-function applied to prove the Riemann conjecture Open
In this article we address the real part of the nontrivial zeros for the Riemann zeta-function by using the new infinite product formulas for the Riemann $\Xi$-function under the latent conditions for the existences of the symmetric real z…
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Series with Binomial-Like Coefficients for the Investigation of Fractal Structures Associated with the Riemann Zeta Function Open
The paper continues the study of efficient algorithms for the computation of zeta functions over the complex plane. We aim to apply the modifications of algorithms to the investigation of underlying fractal structures associated with the R…
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On Euler's Solution of the Simple Difference Equation Open
In this note we will discuss Euler's solution of the simple difference equation that he gave in his paper{\\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \\cite{E189} (E189:``On the determinatio…
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Jacob's ladders and grafting of the complete hybrid formulas into $\zeta$-synergetic meta-functional equation for Riemann's zeta-function. Open
In this paper we prove that there is a grafting of complete hybrid formula on meta-functional equation. This synergetic formula gives the result of a $\zeta$-chemical reaction between different sets of values of Riemann's zeta-function on …
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Explorations in the theory of partition zeta functions Open
We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the …
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Fast convergence of generalized DeTemple sequences and the relation to the Riemann zeta function Open
In this paper, we introduce new sequences, which generalize the celebrated DeTemple sequence, having enhanced speed of convergence. We also give a new representation for Euler's constant in terms of the Riemann zeta function evaluated at p…
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Computer Program in C Programing Language for Calculating the Value of Euler Product equal to the Riemann Zeta Function Open
This paper presents a novel idea to compute the product of geometric series with prime numbers. The product of geometric series with prime numbers plays a vital in the sum of natural numbers, harmonic series, and the Euler product equal to…
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Some Identities and inequalities related to the Riemann zeta function Open
A new proof of Euler’s formular for calculating ζ(2 k ) is given. Some new inequalities and identities for ζ(2 k + 1) have also been given. The Riemann’s functional equation together with trigonometric identities were used to establish the…