![]() The vast experience with the Riemann zeta function in the past and the progress in numerical calculations of the zeros (see, e.g., ) which all confirmed the Riemann hypothesis suggest that it should be true corresponding to the opinion of most of the specialists in this field but not of all specialists (arguments for doubt are discussed in ). The Riemann hypothesis was taken by Hilbert as the 8-th problem in his representation of 23 fundamental unsolved problems in pure mathematics and axiomatic physics in a lecture hold on 8 August in 1900 at the Second Congress of Mathematicians in Paris. Riemann has put aside the search for a proof of his hypothesis “after some fleeting vain attempts” and emphasizes that “it is not necessary for the immediate objections of his investigations” (see ). The book of Borwein, Choi, Rooney and Weirathmueller gives on the first 90 pages a short account about achievements concerning the Riemann hypothesis and its consequences for number theory and on the following about 400 pages it reprints important original papers and expert witnesses in the field. 17) about number analysis, and Apostol (chaps. There are also mathematical tables and chapters in works about Special functions which contain information about the Riemann zeta function and about number analysis, e.g., Whittaker and Watson (chap. The book of Edwards is one of the best older sources concerning most problems connected with the Riemann zeta function. ![]() (with extensive lists of references in some of the cited sources, e.g., ( ). In the complex plane and in extension that all zeros are simple zeros. That means on the line parallel to the imaginary axis through real value The Riemann hypothesis is the conjecture that all nontrivial zeros of the Riemann zeta function for complex are positioned on the line The proof of the Riemann hypothesis is a longstanding problem since it was formulated by Riemann in 1859. The Riemann zeta function which basically was known already to Euler establishes the most important link between number theory and analysis. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions belong also to this case. This whole class includes, in particular, also the modified Bessel functions for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. ![]() All this means that we prove a theorem for zeros of on the imaginary axis z=iy for a whole class of function which includes in this way the proof of the Riemann hypothesis. Besides this theorem we apply the Cauchy- Riemann differential equation in an integrated operator form derived in the Appendix B. Then after a trivial argument displaceme nt we relate it to a function with a representation of t he form where is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy by the second mean-value theorem. The Riemann zeta functio n as it is known can be related to an entire function with the same non-trivial zeros as. The American Mathematical Monthly.By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functions with an integral representation of the form with a real-valued function which is non-increasing and decreases in infinity more rapidly than any exponential functions, possesses zeros only on the imaginary axis. "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" (PDF). "A translation of Bolzano's paper on the intermediate value theorem". Sources and Studies in the History of Mathematics and Physical Sciences. Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction. "The legitimation of geometrical procedures before 1590". "Nonstandard Analysis and Constructivism!". ^ Essentially follows Clarke, Douglas A.Hairy ball theorem – Theorem in differential topology. ![]() Mean value theorem – On the existence of a tangent to an arc parallel to the line through its endpoints.The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily met constraints). This captures an intuitive property of continuous functions over the real numbers: given f -dimensional shape and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same.
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