The Divisor Function in Arithmetic Progressions to Smooth Moduli
نویسندگان
چکیده
منابع مشابه
The divisor function over arithmetic progressions
provided x is sufficiently large. An asymptotic formula of type (1) Df (x; q, a) = (1 +O((log x)))Df (x; q) , in which the error term is smaller than the main term by a suitable power of log x, is good enough for basic applications. More important than the size of the error term is the range where (1) holds uniformly with respect to the modulus q. In this paper we consider the problem for the d...
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where <p is Euler's function and n(x) counts all primes up to x. Indeed, (1.1) is known to hold, but only under the proviso that q grow more slowly than some fixed power of logx (Siegel-Walfisz theorem). Yet, for many applications, it is important to weaken this restriction. The assumption of the Generalized Riemann Hypothesis (GRH) yields (1.1) in the much wider range q < X I / 2/ log2+e x . I...
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We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that “on average” these sums are close to the expected value. We also give applications of our result to sums of the divisor function twisted with characters (both additive and multiplicative) taken on the values of various functions, such as rational and exponential functions; in pa...
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where <p is Euler's function and n(x) counts all primes up to x. Indeed, (1.1) is known to hold, but only under the proviso that q grow more slowly than some fixed power of logx (Siegel-Walfisz theorem). Yet, for many applications, it is important to weaken this restriction. The assumption of the Generalized Riemann Hypothesis (GRH) yields (1.1) in the much wider range q < X I / 2/ log2+e x . I...
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ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2014
ISSN: 1073-7928,1687-0247
DOI: 10.1093/imrn/rnu149