(Edits, adds Paris protest, background)
By Ayla Jean Yackley
ISTANBUL/PARIS, Jan 12 (Reuters) - Turkish Prime Minister Tayyip Erdogan demanded on Saturday that French President Francois Hollande explain why he had met one of three Kurdish militants shot dead in Paris this week.
Kurdish activists blamed the execution-style killings at an institute in central Paris on Thursday on shadowy elements from the Turkish state or foreign powers. Turkey pointed to possible infighting in the separatist Kurdistan Workers Party (PKK).
Speaking to reporters on Thursday, Hollande said one of the three dead women, who all had ties to the PKK, "was known to me and to many politicians because she came to meet us regularly".
The PKK is viewed as a terrorist group by Turkey, the United States and the European Union.
"How can you routinely meet with members of an organisation labelled a terrorist group by the European Union and being sought by Interpol?" Erdogan asked in a speech to a business group broadcast by CNN Turk. "What kind of politics is this?"
"The French president should immediately disclose to the public why he met with members of this terrorist organisation, what was discussed, to what end he was in communication with these terrorists," Erdogan said, adding that Turkey would take unspecified legal action on the matter.
Ties between Ankara and Paris suffered when former French president Nicolas Sarkozy joined forces with Germany's Angela Merkel to resist Turkey's years-long bid to enter the European Union.
Relations remain tense and Hollande has stopped short of endorsing Turkey's candidacy, saying it should be judged on political and economic criteria.
The 28-year-old Kurdish insurgency has held back Turkey's economic and democratic progress.
In an attempt to bring peace, Erdogan's party has expanded political and cultural rights for the country's estimated 15 million Kurds since it came to power in 2002.
Turkish media reported this week that a framework for a new peace plan had been agreed with PKK leader Abdullah Ocalan.
But Kurdish politicians and activists say Erdogan has ensured continued tensions through the arrests and trials of thousands of Kurdish journalists, lawyers and others who have sought a civilian, non-violent resolution of the conflict.
His government has stepped up military attacks on the PKK, including land and air incursions into Iraq to target Kurdish rebels in their mountain hideouts.
NO CLEAR SUSPECTS
French investigators have given no indication as to who might be responsible for the killings in Paris.
One of the dead was Sakine Cansiz, a founding member of the PKK well-known to Kurdish nationalists who was believed to be a key financier in Europe for the group, which was based mainly in northern Iraq.
About 10,000 Kurd supporters gathered in Paris on Saturday to protest at what some have called a politically motivated killing, chanting slogans such as "We are all the PKK."
Erdogan said the killings may be the result of PKK infighting or an attempt to derail the latest peace initiative.
"The killings in Paris may have been an attempt aimed at sabotaging this initiative. It may also be score-settling within the ranks of the separatist terrorist group," he said.
He rejected the allegations that the Turkish state was behind the attack and demanded French authorities apprehend the killers and shed light on the incident.
One pro-Kurd protester in Paris who gave her name as Fintoz said the killings did not stem from a split within the PKK, and accused the Turkish and French governments of "covering up a crime". (Reporting by Ayla Jean Yackley in Istanbul and Noemie Olive in Paris; Editing by Andrew Roche and Tom Pfeiffer)