Geodesic
Distribution in the mean of arithmetic functions in short intervals in progressions
Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 315-335.

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It is shown that arithmetic functions of a certain class including, in particular, the functions $\Lambda(n)$, $\mu(n)$, and $\tau_r(n)$, on the intervals $x$, $y>x^{7/12}$, are uniformly distributed in progressions. The result for $\Lambda(n)$ is as follows. Let $$ \delta(Q,x,y)=\sum_{k\leqslant Q}\max_{(a,k)=1}\max_{\frac x2\leqslant N\leqslant x}\max_{h\leqslant y}\Bigg|\sum_{\substack{N\leqslant N+h\\n\equiv a (\operatorname{mod}k)}}\Lambda(n)-\frac h{\varphi(k)}\Bigg|. $$ Then for $x^{3/5}(\log x)^{2(A+64)+1}\leqslant y\leqslant x$ and $Q=yx^{-1/2}(\log x)^{-(A+64)}$ we have $\delta(Q,x,y)\ll y\log^{-A}x$. If $x^{7/12}$ then this estimate holds, but with $Q=yx^{-11/20-\delta}$, $\delta>0$. Bibliography: 16 titles. 
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N. M. Timofeev. Distribution in the mean of arithmetic functions in short intervals in progressions. Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 315-335. http://geodesic.mathdoc.fr/item/IM2_1988_30_2_a6/

[1] Hohiesel G., “Primzahlprobleme in der Analysis”, Sitz Preuss. Akad. Wiss., 33 (1930), 3–11

[2] Heilbron H., “Über den Primzahlsatz von Herrn Hoheisel”, Math. Z., 36 (1933), 394–423 | DOI | MR | Zbl

[3] Chudakov N. G., “On the difference between two neighboring prime numbers”, Matem. sb., 1 (1936), 799–814

[4] Ingham A. E., “On the difference between consecutive primes”, Quart. J. Math. Oxford., 8 (1937), 255–266 | DOI | Zbl

[5] Montgomery H. L., Zerous of $L$-functions, Invent. Math., 81, 1969 | DOI | MR

[6] Huxley M. N., “On the difference between consecutive primes”, Invent. Math., 15 (1972), 164–170 | DOI | MR | Zbl

[7] Motohashi Y., “On the summ of Mobius function in short segment”, Proc. Japan. Acad., 52:9 (1976), 477–479 | DOI | MR | Zbl

[8] Rama Chandra K., “Some problems of analytic number theory”, Acta Arithm., 34:4 (1976), 313–324

[9] Jutila M., “A statistical density theorem for $L$-functions with applications”, Acta Arithm., 16 (1969), 207–216 | MR | Zbl

[10] Huxley M. N., Iwaniec H., “Bombieri's theorem in short intervals”, Mathematika, 22 (1975), 188–197 | MR

[11] Ricci S. J., “Mean-value theorems for primes in short intervals”, Proc. Lond. Math. Soc., 37:3 (1978), 230–242 | DOI | MR | Zbl

[12] Heath-Brawn D. R., “Prime numbers in short intervals and a generalized Vaughan indentity”, Can. J. Math., 34:6 (1982), 1365–1377 | MR

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

[14] Huxley M. N., “Large values of Dirichlet polynomials, III”, Acta Arithm., 24 (1974), 431–440 | MR

[15] Prakhar K., Raspredelenie prostykh chisel, Mir, M., 1967 | MR

[16] Titchmarsh E. K., Teoriya dzeta-funktsii Rimana, IL, M., 1953