|Nemtsov funeral attracted 50,000 mourners|
He said the most serious attention should be paid to high-profile crimes.
The former deputy prime minister, 55, was murdered on Friday night and buried in Moscow on Tuesday.
The motive is unknown, but Mr Putin's aides have rejected suggestions that he had any involvement. Could they possibly have done otherwise?
Mr Nemtsov, who had been planning a march against the conflict in eastern Ukraine, said recently that he feared the president would have him killed because of his opposition to the war.
He was shot four times in the back while walking with his Ukrainian girlfriend, Anna Duritskaya, on Great Moskvoretsky Bridge.
At least 50,000 people turned out on Sunday to march in tribute to him.
|Vladimir Putin at an interior ministry board meeting in Moscow (4 March)|
The Russian president has vowed to bring the killers to justice
"It is necessary to finally rid Russia of the shame and tragedies like the one that we lived through and saw quite recently. I mean the murder, the brazen murder of Boris Nemtsov right in the centre of the capital," Mr Putin said in televised comments to the interior ministry.
After the shooting, which the Kremlin described as a "provocation" aimed at discrediting the president, Mr Putin said he would do all he could to ensure the killers were brought to justice but little progress appears to have been made in the investigation.
Of course, it would help if you could actually believe anything Mr Putin says, but he is, in my mind, in a class with Goodluck Jonathon, the Nigerian President, whom I doubt ever spoke a truthful word in his life.
When asked by reporters on Tuesday if there were any suspects in the murder, the head of Russia's FSB security service, Alexander Bortnikov, said: "There are always suspects."
|After three days of questioning by Russian authorities, Nemtsov's girlfriend, |
the lovely but clearly shaken Anna Duritskaya flew home to Kyiv yesterday
Opposition leader Alexei Navalny said it was either a government or pro-government organisation. Rogue elements in the security services or fighters returning from eastern Ukraine may have wanted to silence his anti-war stance.
Or it may have been unrelated to Ukraine: Mr Navalny said it could have been ordered by officials in Yaroslavl, where Mr Nemtsov had been investigating corruption.
Obviously, Mr Putin doesn't think this is likely or he wouldn't have used the word "political" in describing the murder.
|Boris Nemtsov's last known movements|
Russia's finance ministry became part of the story on Tuesday when it was asked about reports that a light-coloured car used by the ministry had been seen in the area at the time of the shooting.
The ministry's press service said the Ford car belonged to an in-house security service, but not the ministry itself, Tass news agency reported.
And Lifenews website, which has close links to the security services, quoted the driver, Dmitry Karmaza, as saying he had driven past the scene a few minutes after the shooting, when a patrol car was already there.
Separately, a video purportedly showing a far-right Russian group active in eastern Ukraine claiming it carried out the killing was dismissed by the group's leader, Alexei Milchakov, as a fake.
"We would have liked to bump him off but we wouldn't even have the money for the car," he wrote earlier on social media. Now why couldn't everyone be that honest?