Number Theory

Number theory and its applications: proceedings of a summer by Serguei Stepanov, C.Y. Yildirim

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By Serguei Stepanov, C.Y. Yildirim

Addresses modern advancements in quantity conception and coding idea, initially provided as lectures at summer time tuition held at Bilkent collage, Ankara, Turkey. comprises many leads to e-book shape for the 1st time.

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E. up to height T for some explicit, large T ) of an L-function lie on the critical line seems at first sight to be strong evidence for the corresponding RH. However, analytic number theory has had many conjectures supported by large amounts of numerical evidence that turn out to be false. , [Iv18]). The problem is that the behavior of a function is often influenced by very slowly increasing functions such as log log T , that tend to infinity, but do it so slowly that this cannot be detected by computation.

47), see K. Soundararajan [Sou4]. He proved that one can take C = 3/8. A sharper result, also on the RH, was obtained by V. Chandee and K. Soundararajan [ChSo], namely |ζ ( 12 + it)| log 2 log t 2 log log t exp 1+O log log log t log log t . 53) as shown by K. Soundararajan [Sou3]. 33) explicit. Namely it was proved by T. S. 17 log T . This is an unconditional result. On the RH, K. Ramachandra and A. 2 log T log log T (T > T0 ). This was improved, √ again on the RH, by D. A. Goldston and S. M. Gonek [GoGo].

It may be conjectured that both S(T ) and S1 (T ) are of the order (log T )1/2+o(1) as T → ∞, although it is known, for example, only that S(T ) = O(log T ) (and O(log T / log log T ) if the RH holds), so that there is still a considerable gap between O- and -results. -M. Tsang [Tsa2] has shown that unconditionally sup log |ζ ( 12 + it)| sup ±S(t) T

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