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The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.
The Riemann zeta function is an example of an L-function, and some important conjectures involving L-functions are the Riemann hypothesis and its generalizations. The theory of L -functions has become a very substantial, and still largely conjectural , part of contemporary analytic number theory .
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function . Various geometrical and arithmetical objects can be described by so-called global L -functions , which are formally similar to the Riemann zeta-function.
In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann , is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields .
Psychological statistics is application of formulas, theorems, numbers and laws to psychology. Statistical methods for psychology include development and application statistical theory and methods for modeling psychological data. These methods include psychometrics, factor analysis, experimental designs, and Bayesian statistics. The article ...
the Riemann hypothesis is equivalent to the statement that > for every positive integer . The numbers (sometimes defined with a slightly different normalization) are called Keiper-Li coefficients or Li coefficients. They may also be expressed in terms of the non-trivial zeros of the Riemann zeta function:
Riesz showed that the Riemann hypothesis is equivalent to the claim that the above is true for any e larger than /. [1] In the same paper, he added a slightly pessimistic note too: « Je ne sais pas encore decider si cette condition facilitera la vérification de l'hypothèse » ("I can't decide if this condition will facilitate the ...