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High Quality Content by WIKIPEDIA articles! In biochemistry and molecular biology, a residue refers to a specific monomer within the polymeric chain of a polysaccharide, protein or nucleic acid. For example, one might say, "The histidine residue is considered to be basic due to its imidazole ring." Note that a residue is different from a moiety, which, in the above example would be constituted by the imidazole ring or "the imidazole moiety".Note the origin of this usage: during the process by which monomeric building blocks (e.g. amino acids) are strung together into a polymeric chain (e.g. a protein), some material (typically adding up to one molecule of water) is discarded from each building block, and only a "residue" of the building block ends up in the finished product.For example, a residue is an individual amino acid in a peptide chain. ... Read more

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High Quality Content by WIKIPEDIA articles! A mathematical game is a multiplayer game whose rules, strategies, and outcomes can be studied and explained by mathematics. Examples of such games are Tic-tac-toe and Dots and Boxes, to name a couple. On the surface, a game need not seem mathematical or complicated to still be a mathematical game. For example, even though the rules ofMancala are straightforward, mathematicians analyze the game using combinatorial game theory. Mathematical games differ from mathematical puzzles in that all mathematical puzzles require math to solve them whereas mathematical games may not require a knowledge of mathematics to play them or even to win them. Thus the actual mathematics of mathematical games may not be apparent to the average player. Some mathematical games are topics of interest in recreational mathematics. ... Read more

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Riemann Hypothesis. Bernhard Riemann, Riemann Zeta Function, Conjecture, Hilbert?Palya Conjecture, Generalized Riemann Hypothesis, Lee?Yang Theorem, Local Zeta-Function, Explicit Formula, Logarithmic Integral Function, Leonhard Euler, Euler Product, On the Number of Primes Less Than a Given Magnitude, Article Sources and Contributors, Image Sources, Licenses and Contributors ... Read more

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Prime Number. Natural Number, Divisor, Prime Number Theorem, Primality Test, Largest known Prime Number, Mersenne Prime, Formula for Primes, Prime-Counting Function, Euclid's Theorem, Prime Gap, Riemann Hypothesis, Modular Arithmetic, Integral Domain, Prime Power, Prime Ideal, Fundamental Theorem of Arithmetic, Article Sources and Contributors, Image Sources, Licenses and Contributors ... Read more

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These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. In 1975, Voronin proved that any non-vanishing analytic function can be approximated uniformly by certain shifts of the Riemann zeta-function in the critical strip. This spectacular universality property has a strong impact on the zero-distribution: Riemann’s hypothesis is true if and only if the Riemann zeta-function can approximate itself uniformly (in the sense of Voronin). Meanwhile universality is proved for a large zoo of Dirichlet series, and it is conjectured that all reasonable L-functions are universal. In these notes we prove universality for polynomial Euler products. Our approach follows mainly Bagchi's probabilistic method. We further discuss related topics as, e.g., almost periodicity, density estimates, Nevanlinna theory, and functional independence.

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Number theory, spectral geometry, and fractal geometry are interlinked in this study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. The Riemann hypothesis is given a natural geometric reformulation in context of vibrating fractal strings, and the book offers explicit formulas extended to apply to the geometric, spectral and dynamic zeta functions associated with a fractal.

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"Excerpt from the book..."

In venturing to prepare this little volume for the eyes of the reading
public, I am fully aware of the difficulties of the subject and the
inadequacy of the expressions I have been able to employ, but I have
made the attempt at the request of those who have found consolation in
some of the thoughts herein embodied
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