Geodesic
On the distribution of the values of $L(1,\chi_{8p})$
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 253-273 Cet article a éte moissonné depuis la source Math-Net.Ru

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The moments of pure imaginary and integer orders of the function $L(1,\chi_{8p})$, where $\chi_{8p}(n)=(8p/n)$ and $p$ runs over all primes $p>2$, are computed. In order to derive uniform variants of the theorems on moments, the extended Riemann hypothesis for the Dirichlet $L$-series must be used. As corollaries, the limiting distribution of the values of $\log L(1,\chi_{8p})$ is studied, and quantitative analogs of the $\Omega$-results for $L(1,\chi_{8p})$ are obtained. Previously, $\Omega$-results for $L(1,\chi_p)$ were proved by Bateman, Chowla, and Erdos (1949–1950) and by Barban (1966), and their methods can easily be transferred to $L(1,\chi_{8p})$. Bibliography: 27 titles. 
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O. M. Fomenko. On the distribution of the values of $L(1,\chi_{8p})$. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 253-273. http://geodesic.mathdoc.fr/item/ZNSL_2006_337_a14/

[1] J. E. Littlewood, “On the class-number of the corpus $P\sqrt{-k}$”, Proc. London Math. Soc. (2), 27 (1928), 358–372 | DOI | MR | Zbl

[2] S. Chowla, “On the class-number of the corpus $P\sqrt{-k}$”, Proc. Nat. Acad. Sci. India, 13 (1947), 197–200 | MR

[3] S. Chowla, “Improvement of a theorem of Linnik and Walfish”, Proc. London Math. Soc. (2), 50 (1949), 423–429 | DOI | MR

[4] P. T. Bateman, S. Chowla, P. Erdös, “Remarks on the size of $L(1,\chi)$”, Publ. Math. Debrecen, 1 (1949–50), 165–182 | MR

[5] M. B. Barban, “Metod “bolshogo resheta” i ego primeneniya v teorii chisel”, Uspekhi mat. nauk, 21:1 (1966), 51–102 | MR | Zbl

[6] M. B. Barban, ““Bolshoe resheto” Yu. V. Linnika i predelnaya teorema dlya chisla klassov idealov mnimogo kvadratichnogo polya”, Izv. AN SSSR. Ser. mat., 26:4 (1962), 573–580 | MR | Zbl

[7] A. S. Fainleib, “O raspredelenii chisla klassov polozhitelnykh kvadratichnykh form”, Nauch. tr. Tashkent. un-t, 418, Tashkent, 1972, 272–279 | MR | Zbl

[8] G. Montgomeri, Multiplikativnaya teoriya chisel, Mir, M., 1974 | MR

[9] A. Ivić, The Riemann zeta-function, Wiley, New York etc., 1985 | MR | Zbl

[10] E. P. Golubeva, “Raspredelenie znachenii $L$-funktsii Gekke v tochke $1$”, Zap. nauchn. semin. POMI, 314, 2004, 15–32 | Zbl

[11] O. M. Fomenko, “O raspredelenii znachenii $L(1,\mathrm{sym}^2f)$”, Algebra i analiz, 17:6 (2005), 184–206 | MR

[12] C. Hooley, “On the Barban–Davenport–Halberstam theorem, I”, J. Reine Angew. Math., 274–275 (1975), 206–223 | DOI | MR | Zbl

[13] K. K. Mardzhanishvili, “Otsenka odnoi arifmeticheskoi summy”, Dokl. AN SSSR, 22:7 (1939), 391–393

[14] P. D. T. A. Elliott, “On the mean value of $f(p)$”, Proc. London Math. Soc. (3), 21 (1970), 28–96 | DOI | MR | Zbl

[15] P. D. T. A. Elliott, “The distribution of the quadratic class number”, Lit. mat. sb., 10:1 (1970), 189–197 | MR | Zbl

[16] A. C. Berry, “The accuracy of the Gaussian approximation to the sum of independent variables”, Trans. Amer. Math. Soc., 49 (1941), 122–136 | DOI | MR

[17] C. G. Esseen, “Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law”, Acta Math., 77 (1945), 1–125 | DOI | MR | Zbl

[18] A. S. Fainleib, “Obobschenie neravenstva Esseena i ego primenenie v veroyatnostnoi teorii chisel”, Izv. AN SSSR. Ser. mat., 32:4 (1968), 859–879 | MR | Zbl

[19] Yu. V. Linnik, Razlozhenie veroyatnostnykh zakonov, Izd-vo Leningr. un-ta, L., 1960 | MR

[20] H. L. Montgomery, R. C. Vaughan, “Extreme values of Dirichlet $L$-functions at $1$”, Number Theory in Progress, Vol. 2 (Zakopane, Poland, 1997), Gruyter, Berlin etc., 1999, 1039–1052 | MR | Zbl

[21] E. P. Golubeva, “Raspredelenie znachenii $L$-funktsii Dirikhle v $1$” (to appear)

[22] P. D. T. A. Elliott, “On the distribution of the values of quadratic $L$-series in the half-plane $\sigma>\frac12$”, Invent. Math., 21 (1973), 319–338 | DOI | MR | Zbl

[23] A. I. Vinogradov, “O nulyakh Zigelya”, Dokl. AN SSSR, 151:3 (1963), 479–481 | MR | Zbl

[24] E. Royer, “Interprétation combinatoire des moments négatifs des valeurs de fonctions $L$ au bord de la bande critique”, Ann. Sci. École Norm. Sup. (4), 36:4 (2003), 601–620 | MR | Zbl

[25] A. F. Lavrik, “Metod otsenok dvoinykh summ s veschestvennym kvadratichnym kharakterom i ego prilozheniya”, Izv. AN SSSR. Ser. mat., 35:6 (1971), 1189–1207 | MR | Zbl

[26] R. C. Vaughan, “Small values of Dirichlet $L$-functions at $1$”, Anatytic Number Theory, Vol. 2 (Allerton Park, IL, 1995), Prog. Math., 139, Birkhäuser Boston, Boston, MA, 1996, 755–766 | MR | Zbl

[27] E. Stankus, “Elementarnyi metod otsenki kompleksnykh momentov $L(1$,$\chi_D)$”, Lit. mat. sb., 22:2 (1982), 160–170 | MR | Zbl