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V magistrskem delu obravnavamo Bertrandovo domnevo, ki pravi, da za vsako naravno število $n$ obstaja vsaj eno praštevilo $p$, za katerega velja $n < p leq 2n$. Podrobneje predstavimo nekaj najbolj znanih dokazov Bertrandove domneve - Erdősev, Ramanujanov in poenostavljen Ramanujanov dokaz. Erdősev dokaz temelji na oceni binomskega koeficienta, Ramanujanov dokaz pa izhaja iz prvega dokaza Bertrandove domneve, Čebiševega dokaza iz leta 1852. Poenostavljen Ramanujanov dokaz sta zapisala avtorja Meher in Ram Murty, ki sta domnevo dokazovala na enak način kot Ramanujan, le da sta se izognila uporabi Stirlingove formule. Opišemo tudi nekaj modifikacij Bertrandove domneve in njihovih uporab, med drugim Ramanujanova praštevila, ki jih Ramanujan uvede na koncu dokaza Bertrandove domneve. In the master's thesis, we study Bertrand's postulate, which states that for any given natural number $n$, there exists at least one prime number $p$ such that $n < p leq 2n$. We present in more detail some of the most famous proofs of Bertrand's postulate - Erdős's, Ramanujan's and simplified Ramanujan's proof. Erdős's proof is based on estimation of the binomial coefficient. Ramanujan's proof derives from the first proof of Bertrand's postulate, Chebyshev's proof from 1852. Ramanujan's simplified proof was written by Meher and Ram Murty, who proved the postulate in the same way as Ramanujan, except they avoided using Stirling's formula in their proof. We also describe modifications of Bertrand's postulate and their applications, including Ramanujan's primes that Ramanujan introduces at the end of the proof of Bertrand's postulate.

V delu z analitičnimi metodami dokažemo praštevilski izrek. V ta namen predstavimo osnovno teorijo neskončnih produktov in vpeljemo Riemannovo funkcijo zeta. Izpeljemo Eulerjevo produktno formulo, poiščemo meromorfno razširitev funkcije zeta na desno polovico kompleksne ravnine in predpis za njen logaritmični odvod. Definiramo Mangoldtovo funkcijo in funkcijo psi ter z njuno pomočjo poiščemo ekvivalentno obliko praštevilskega izreka, ki ga nazadnje dokažemo z metodami kompleksne analize. In this work, prime number theorem is proven using analytic methods. For this purpose elementary theory of infinite products is introduced and the Riemann zeta function is used. We derive the Euler product formula and find a meromorphic extension of the zeta function to the right half of the complex plane and the expression for its logarithmic derivative. We also define the Mangoldt and psi function and use them to find an equivalent formulation of the prime number theorem. Finally, the prime number theorem is proved using complex analytic methods.

Riemannovo funkcijo zeta definiramo kot funkcijo kompleksne spremenljivke $s$, in sicer z vrsto $zeta(s) = sum_{n = 1}^{infty} frac{1}{n^s}$ za $operatorname{Re}s > 1 $, nato pa jo analitično razširimo na ${mathbb C} setminus {1}$. Pri tem si pomagamo s funkcijsko enačbo, v kateri je pomembna vloga funkcije gama. Tako razširjena funkcija zeta ima v točki 1 pol stopnje 1, v točkah $-2, -4, -6, dots$ pa tako imenovane trivialne ničle. V nadaljavanju Riemannovo funkcijo zeta izrazimo kot neskončen produkt, imenovan Eulerjev produkt, in pokažemo, da $zeta$ nima ničel na polravnini $operatorname{Re}s geq 1$. To dejstvo uporabimo v dokazu praštevilskega izreka, ki govori o asimptotični ekvivalenci funkcij $pi (x)$ in $x / ln(x)$, kjer s $pi (x) $ označimo število praštevil, ki so manjša ali enaka danemu pozitivnemu realnemu številu $x$. We define the Riemann zeta function as a function of a complex variable $s$ with the series $zeta(s) = sum_{n=1}^{infty} frac{1}{n^s}$ for $operatorname{Re}s > 1 $ and then extend it analytically to ${mathbb C} setminus {1}$. We use a functional equation in which the gamma function plays an important role. The extended zeta function has a simple pole in 1 and so-called trivial zeros in $-2, -4, -6, dots$. Later on, we express the Riemann zeta function as an infinite product called the Euler product and show that $zeta$ has no zeros on the half-plane $operatorname{Re} s geq 1$. We use this fact in the proof of the prime number theorem which describes the asymptotic equivalence of the functions $pi (x)$ and $x / ln (x) $, where $pi (x)$ denotes the number of primes less than or equal to a given positive real number $x$.