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In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the ...
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2 , and results in 9, that is, 1 + 2 + 4 + 2 = 9 . Because addition is associative and commutative , there is no need for parentheses, and the result is the same irrespective of the order of the summands.
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
Von Mangoldt provided a rigorous proof of an explicit formula for ψ(x) involving a sum over the non-trivial zeros of the Riemann zeta function. This was an important part of the first proof of the prime number theorem. The Mellin transform of the Chebyshev function can be found by applying Perron's formula:
Inclusion–exclusion principle. In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as. where A and B are two finite sets and | S | indicates the cardinality of a ...
The Baer sum is an explicit description of the abelian group structure on Ext 1 R ( A , B ), viewed as the set of equivalence classes of extensions of A by B . [ 10 ] Namely, given two extensions
A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If + + + =, then adding 0 to both sides gives
Hilbert's proof did not exhibit any explicit counterexample: only in 1967 the first explicit counterexample was constructed by Motzkin. [3] Furthermore, if the polynomial has a degree 2d greater than two, there are significantly many more non-negative polynomials that cannot be expressed as sums of squares. [4]