Mathematics PhD Thesis Defense by Tavakgul Mehraliyev

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KOÇ UNIVERSITY

GRADUATE SCHOOL OF SCIENCES & ENGINEERING

MATHEMATICS

PhD THESIS DEFENSE BY TAVAKGUL MEHRALIYEV

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Title: Number of Prime Ideals in Short Intervals

Speaker: Tavakgul Mehraliyev

Time: September 12, 2017,  15.00

Place: ENG 208

Koç University

Rumeli Feneri Yolu

Sariyer, Istanbul

Thesis Committee Members:

Assoc. Prof. Emre Alkan (Advisor, Koç University)

Prof. Özgür Müstecaplıoğlu (Koç University)

Assoc. Prof. Sinan Ünver (Koç University)

Asst. Prof. Mehmet Öz (Özyeğin University)

Asst. Prof. Cihan Pehlivan (Atılım University)

Abstract:

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun-Titchmarsh theorem.