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...al hierarchy]]); these theorems are studied as part of [[hyperarithmetical theory]]. ...ow that there must be points that are not "too far" from being computable, in an informal sense. ...

In [[analytic number theory]], the '''Petersson trace formula''' is a kind of orthogonality relation be In its simplest form the Petersson trace formula is as follows. Let <math>\mat ...

...]] on the distribution of [[primes in arithmetic progression|prime numbers in arithmetic progression]]. Let <math>\pi(x;q,a)</math> count the number of primes ''p'' congruent to ''a'' modulo&nbsp;''q'' with ''p''&nbsp;≤&nbsp ...

...ematics]], '''Wilkie's theorem''' is a result by [[Alex Wilkie]] about the theory of [[ordered field]]s with an [[Exponential field|exponential function]], o ...>1</sub>, ..., ''x''_''m'') is a [[well-formed formula|formula]] in this language. Then Wilkie's theorem states that there is an [[integer]] '' ...

...ries is "badly behaved" in the sense that it cannot be extended to be an [[analytic function]] anywhere on the [[boundary (topology)|boundary]] of its [[radius :<math>f(z) = \sum_{j \in \mathbf{N}} \alpha_{j} z^{p_{j}}</math> ...

...istributed equally across possible progressions with the same difference. Theorems of the Barban–Davenport–Halberstam type give estimates for the error term, be a weighted count of primes in the arithmetic progression ''a''&nbsp;mod&nbsp;''q''. We have ...

...series related to [[Lambert series]] specially relevant in analytic number theory. ...rt summable to <math>A</math>. Then it is Abel summable to <math>A</math>. In particular, if <math>\sum_{n=0}^\infty a_n</math> is Lambert summable to <m ...

{{Short description|Term used in transcendental number theory}} ...971514 | jstor=1971514 | mr=997310}}</ref> It marked a breakthrough in the theory of transcendental numbers. Many longstanding open problems can be deduced a ...

In [[analytic number theory]], the '''Kuznetsov trace formula''' is an extension of the [[Petersson tra ...'' formula connects [[Kloosterman sum]]s at a deep level with the spectral theory of [[automorphic form]]s. Originally this could have been stated as follows ...

...lm Wirtinger]] in 1932 in connection with some problems of [[approximation theory]]. This theorem gives the representation formula for the [[Holomorphic func }}</ref> contains the following theorem presented also in [[Joseph L. Walsh]]'s well-known monograph ...

...eorem due to [[Helmut Maier]] about the numbers of [[Prime number|prime]]s in short intervals for which [[Cramér model|Cramér's probabilistic model of pr ...math> u </math> fixed). He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to [[Patrick X. Gallagher| ...

In the field of [[mathematical analysis]], a '''general Dirichlet series''' is where <math>a_n</math>, <math>s</math> are [[complex number]]s and <math>\{\lambda_n\}</math> is a strictly increasing [[sequence]] of ...

...2000 as Regents Professor, and from 2003 as Vaughan Foundation Professor). In 2004 he became a professor at [[Ohio State University]].<ref name=homepage> In autumn 1983 and for the academic year 1999–2000 he was at the [[Institute f ...

{{About|a theorem in complex analysis||Hurwitz's theorem (disambiguation){{!}}Hurwitz's theorem} In [[mathematics]] and in particular the field of [[complex analysis]], '''Hurwitz's theorem''' is a ...

In [[number theory]] and [[harmonic analysis]], the '''Landsberg–Schaar relation''' (or '''ide \frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi in^2q}{p}\right)= ...

{{Short description|Theorem in mathematics}} '''Richards' theorem''' is a mathematical result due to [[Paul I. Richards]] in 1947. The theorem states that for, ...

{{Short description|Concept in algebraic geometry}} In [[mathematics]], especially in [[algebraic geometry]] and the theory of [[complex manifold]]s, '''coherent sheaf cohomology''' is a technique fo ...

{{Short description|Computes the Poincaré–Hopf index of a real, analytic vector field at a singularity}} ...ex. The signature formula expresses the index of an analytic vector field in terms of the [[signature of a quadratic form|signature]] of a certain [[qua ...

...ticular type of sequences corresponding to sums of many terms, are covered in the article on [[convergence tests]]. ==Convergence in '''R'''^''n''== ...

In [[mathematics]], the '''Gross–Koblitz formula''', introduced by {{harvs|txt ...s–Koblitz formula states that the Gauss sum <math>\tau</math> can be given in terms of the <math>p</math>-adic gamma function <math>\Gamma_p</math> by ...