Dirichlet L-functions
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11: 27.9 Quadratic Characters
§27.9 Quadratic Characters
… ►The Legendre symbol , as a function of , is a Dirichlet character (mod ). … ►The Jacobi symbol is a Dirichlet character (mod ). …12: 27.11 Asymptotic Formulas: Partial Sums
§27.11 Asymptotic Formulas: Partial Sums
… ►For example, Dirichlet (1849) proves that for all , …Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number such that the error term in (27.11.2) is for all . … ►where , . ►Letting in (27.11.9) or in (27.11.11) we see that there are infinitely many primes if are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. …13: 24.16 Generalizations
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§24.16(ii) Character Analogs
►Let be a primitive Dirichlet character (see §27.8). Then is called the conductor of . … ►
24.16.11
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24.16.12
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14: 30.14 Wave Equation in Oblate Spheroidal Coordinates
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§30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids
…15: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält.
Abhandlungen der Königlich Preussischen Akademie der
Wissenschaften von 1837, pp. 45–81 (German).
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Über die Bestimmung der mittleren Werthe in der Zahlentheorie.
Abhandlungen der Königlich Preussischen Akademie der
Wissenschaften von 1849, pp. 69–83 (German).
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16: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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§30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids
… ►For the Dirichlet boundary-value problem of the region between two ellipsoids, the eigenvalues are determined from …17: 27.3 Multiplicative Properties
18: 14.12 Integral Representations
19: 25.2 Definition and Expansions
20: Bibliography
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Note on the trivial zeros of Dirichlet
-functions.
Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
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Modular Functions and Dirichlet Series in Number Theory.
2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
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